Spectroscopic Constants and Potential Energy Curves of Heavy p-Block Dimers and Trimers K. BALASUBRAMANIAN' oepert"t
Of
-by.
A r L " State MWtVsiIy, T w , Arllona 85287-1604
Contents I. Introduction 11. Methods of lnvestigatbns A. Experimental Techniques 8. Theoretical Techniques of Caiculatbns 111. Spectroscopic Properties and Potential Energy Curves of Heavy Homonuclear Dimers (Ga, to Biz) A. Gap B. Oe, C. As, D. Se, E. Br, and Er,' F. In, 0. Sn, H. Sb, 1. Te, J. I, and I,+ K. Ti, and T,+ i L. Pb, M. Biz
IV. Spectroscopic Properties and Potential Energy Curves of Heteronuclear Dimers A. GaAs and GaAs+ B. KrEr' and Energy Transfer in &+-Kr Collisions C. IC1 and ICI+ V. Speciroscopic Propertles and Potential Energy Surfaces of Trimers A. Ga, 8. GaAs, C. Ge, and Si, D. In,
E. Sn, VI. Enumeration of the Isomers of Gallium ArsenMe Clusters (Ga,As.) VII. Comparison of the Propertles and Periodic Trends among Dimers VIII. Conclusion IX. Acknowledgments
93 94 94 95 96
96 98 100 102 106 111 115 117 123 127 132 133 135 136 136 140 143 147 147 149 151 151 155 157 161 163 163
I . Introtlucth The electronic and spectroscopic properties of small clusters of heavy atoms and metal atoms have been the 'Camille and Henry Dreyfus Teacher-Scholar.
--_ ,\
.
\ Krlshnan Baiasukamanian has been an Assmiate Professw of CblIWty slna, 1987 at Arkma State UntverSHy. Tempe. AZ. He was born in Bangalore. Indla. in 1956. He received his MSc. (Hawrs)desee from the &a ! 1e of Techndogy and sdence. Pilani. India. in 1977 and his M.A. and Ph.D. degrees from The Johns Hopkims LhiverSny in 1980. He was a pcstdoctaai assodate
and visiting lectuer in ttm Depamnent of Chemishy and Lawrence Berkeley Laboratwy. University of California. Berkeley. from 1980 to 1983. From 1983 10 1986 he was an Assistant Professor at Aruona State University. He received ttm Alfred P. Sloan Fetlowship in 1984 and ttm Camille and Henry Dreytus Teacher-Scholar Award in 1985. Professor Ealasubramanlan's research interests include the relativistic quantum chemistry of molecules containing very heavy atoms. chemical applications of group theory. graph theory. and artificialintelligence. He is an author or coauthor of 175 scientificpublications.
topics of many e~perimental'-'OJ"'~~ and theoretical s t u d i e F 0 in recent years. The intense theoretical and experimental activity in this area has culminated in a wealth of information on the spectroscopic properties and potential energy curves of many heavy dimers and trimers of these species. Investigations of dimers and trimers of such species provide details on the low-lying electronic states, the nature of the heavy atom-atom bond, and periodic trends within various groups. The study of clusters could provide important insights and links to the formation of bulk and solid state from molecular states. Experimental progress in this area has been phenomenal in recent y e a r ~ ~ ~ O due J - " to ~ the advent of laser vaporization and supersonic jet expansion methods.Typically, a sample of a foil or sheet of material containing these elements such as a GaAs c r y ~ t a l ? ~ J ~ a Si crystal,% or Pt foil"' is vaporized with a high-energy Nd:YAG laser and passed through a supersonic nozzle. The jet-cooled beam of clusters with varied compositions is then photoionized and investigated by laser spectroscopic methods. A number of other techniques such as sputtering methods,8?-89rare-gas matrix isolation methods?* particle bombardment meth-
0009-2665/90/0790-0093$09.50/0 0 1990 American
Chemical Society
94 Chemical Reviews, 1990, Vol. 90, No. 1
0ds20J'5J16ligand-stripping methods,w9z and aggregation within zeolites and molecular sievesg3-%have been used for cluster generation. Theoretical ab initio on such clusters are on the increase due to advances both in theoretical methods and in supercomputers that utilize powerful software techniques to produce meaningful numbers and to yield information of significance to experimentalists and of great value in enhancing our knowledge of bonding in these systems. Relativistic effective core potentials (RECP)"lw have become immensely useful and viable alternatives to formidably difficult all-electron calculations for systems as complex as a cluster containing very heavy atoms such as Pb, Sn, Bi, Sb, etc. The RECPs in conjunction with powerful techniques such as the complete active space multiconfiguration self-consistent field (CASSCF) method and configuration interaction calculations (CI)527"have provided reliable results that can be used to either interpret existing experimental information or predict future experiments. Theoretical calculations have also become valuable in designing new experiments. Clusters of heavy main-group elements comprise a subject themselves, a separate class of unusual compounds whose properties exhibit phenomenal variation as a function of size. Many experimental and theoretical investigations in recent years have demonstrated that the geometries and electronic properties of small clusters containing typically 3-10 atoms have no resemblance at all to the more familiar properties of the bulk. It is this aspect that has made the cluster area perhaps one of the most exciting and challenging areas for both experimental and theoretical activities. Electron affinities and ionization potentials of these species exhibit dramatic alternations as a function of cluster size. The stabilities of these clusters as a function of size exhibit certain numbers, for which clusters are unusually more stable, referred to as magic numbers. The existence of magic numbers is quite intriguing and needs explanation. Although a considerable amount of information has been accumulated in recent years on such cluster particles, understanding and full rationalization of all the facts are yet to come. The cluster area is thus relatively young compared to the more established traditional area of main-group inorganic chemistry or organometallic chemistry. The objective of the present review is to organize, catalogue, and comrehend the existing information derived from both experimental and theoretical techniques with the intent of promoting further growth in this developing area of research. Although there is a considerable amount of information on larger clusters, we restrict our review to mainly homonuclear dimers, trimers, and some heteronuclear dimers and trimers of such heavy species. A recent review by Mandich et al.' has placed special emphasis on larger clusters. For most of the heavy main-group dimers, theoretical calculations are now available. Thus, dimers are reviewed here a bit more exhaustively than possible heteronuclear dimers or trimers. Since the emphasis of this review is more on a comparison of theory and experiment, it is restricted to the heavy p-block dimers Gaz to Biz with the exceptions of rare-gas dimers, which mostly form van der Waals complexes in their ground states. The heteronuclear dimers reviewed include GaAs, GaAs+, KrBr+, ICl, and IC]+. The properties of
Balasubramanian
the heavy trimers Ga3,Ge3, Si3,GaAsz,In3, and Sn, are also reviewed. Section I1 briefly outlines both experimental and theoretical techniques employed to investigate small clusters. Section I11 focuses on spectroscopic properties and potential energy curves of Gaz to Biz. Section IV reviews the spectroscopic properties and potential energy curves of some heteronuclear dimers. Section V discusses the properties of small trimers, and section VI discusses the combinatorics of clusters. Section VI1 presents periodic trends among the dimers and trimers reviewed here.
I I. Methods of Investigations A. Experimental Techniques
Mandich et al.' have reviewed in depth a number of experimental techniques employed to generate and probe clusters. In this review, we briefly discuss some of these techniques to make it self-contained. The generation of heavy main-group clusters can be accomplished through a number of methods including evaporation method^,^'-^^ jet expansion method^,^^-^^ sputtering technique^,^^-^^ aggregation meth0ds,9~-~~ rare-gas matrix isolation methods,3zand ligand-stripping methods.w9z In recent years, supersonic jet expansion methods have been employed with much success for cluster g e n e r a t i ~ n . ~ & ~ v ~ ~ - ' ~ ' Simple evaporation of a source material in an oven can generate small cluster particles of heavy group V elements such as Asz, Sbz, Biz, Bi4, and Sb4.71-77The oven evaporation method followed by mass spectrometric analysis can unambiguously determine the cluster size. The use of a Knudsen cell containing an effusion orifice is especially suitable for measuring the binding energies and D,O values of small clusters using the second- or third-law thermodynamic method^.^^-^^ Many group 111-V clusters have been generated by these methods. The main limitation of a simple evaporation method seems to be size; at present, clusters containing at most ten atoms can be generated by this method. Also, this method is not suitable for heteronuclear clusters and alloy clusters. A popular method for generating clusters of both main-group and transition-metal atoms appears to be the supersonic jet expansion m e t h ~ d . ~ " ~ ~In~ this ~-'~' method, typically a high-energy laser evaporation method is used to form the vapor of the material. This is then expanded into a small nozzle, resulting in the formation of clusters of varied composition. This technique can be used to generate simple clusters and mixed clusters of both refractory and nonrefractory materials. Laser vaporization followed by gas aggregation can yield cold beams of clusters that can be probed further by laser spectroscopic methods. Typically, a pulsed Nd:YAG laser is employed in this method. Smalley and c o - w o r k e r ~ ~have " ~ pioneered this technique for many main-group and transition-metal clusters. Clusters containing up to 55 atoms of the main-group elements and carbon clusters in the range C ,-C190 have been generated by these techniques. Mixed main-group clusters such as Ga,As ,97-99 In,P 104~106Sn,Bi,, Pb,Sb,,84J1~11zand Ga,P,"?have also teen generated by the evaporation of 111-V semiconductor and alloy targets.
Heavy p8lock Dimers and Trimers
Bondybey and C O - W O ~ ~ have ~ ~ isolated S ~ ~ clusters of heavy main-group elements such as Pb2, Sn2, etc. using a sputtering method as well as laser vaporization methods. Clusters of various sizes are sputtered and trapped in rare-gas matrices. The matrix-isolated clusters are then probed by laser-induced fluorescence (LIF) methods. For example, Bondybey and EnglishE7 have investigated Pb2 using the sputtering method.87 A disadvantage of this method is that the size of the cluster cannot be determined unambiguously. Cluster formation can be induced via aggregation of atoms on a support such as molecular sieves or a This technique involves simple deposition of elemental vapor onto a target in which the atoms are coalesced into small clusters. Other techniques include the ligand-stripping methodaWg2In this technique, naked clusters are formed by stripping ligands such as CO, H, etc. from a more stable organometallic compound. The stripping is normally induced by high-energy collision with species such as rare-gas atoms, a process that can be called collision-induceddissociation (CID). For example, Asz+can be produced from As2H4 by this method.g0 The electron-impact method can also be used to produce charged clusters. Clusters generated by the above-mentioned methods can be probed by a variety of techniques including laser-induced fluorescence, multiphoton ionization and dissociation spectroscopy, especially two-photon and two-color methods, resonance multiphoton methods, time-of-flight and quadrupole mass spectroscopic methods, ion cyclotron resonance mass spectroscopic methods, Raman spectroscopy, visible-UV electronic spectroscopy, and ESR spectroscopy. The mass spectral methods are mainly for the detection of clusters, calibration of cluster sizes, and measurement of relative abundance as a function of size. The other spectroscopic techniques provide valuable information on the low-lying electronic states, geometries, stabilities, ionization potentials, and electron affinities as a function of size and the spin multiplicities of the low-lying electronic states. The negative-ion photoelectron spectroscopic technique has also provided valuable information on the electronic states of neutral species via ionization of an electron of the anion.34p36J1g121 The photofragmentation patterns of the cationiclmand anionic clusters have also provided some fascinating information that is far from understood. For example, the photofragmentation patterns of Ga,Asy+ clusters" exhibit anomalous patterns that are more reminiscent of metal clusters than of isoelectronic Si, and Ge, clusters.98 Similarly, the photofragmentation patterns of Sb, and Bi, clusters are dramatically different. The Si,+ and Ge,+ clusters fragment into daughters containing 6-11 atoms while metal clusters lose one atom at a time. The photoionization of Ga,Asy semiconductor clusters exhibits dramatic even-odd alternations. B. Theoretical Techniques of Calculations
All the calculations of dimers and trimers reviewed here were done with relativistic effective core potent i a l ~ In. the ~ ~ earlier ~ ~investigations ~ on Pb2,Sn,, Tlz, and Biz, these potentials were used in conjunction with
Chemical Reviews, 1990, Vol. 90, No. 1 95 ~
Slater ~ ~ type ~ ~ orbital ~ ~ (STO) ~ ~ ~basis sets of double-f + polarization quality. The calculations that employed STO basis sets were done mostly by using the single-configuration self-consistent field (SCF) method or a generalized valence bond (GVB) method in the case of the ground state of Biz followed by relativistic configuration interaction (RCI) calculations. The RCI calculations included the spin-orbit term derived from RECPs as suggested by Ermler et alawand Hafner and Schwarz.63 The calculations including the spin-orbit term are called relativistic CI calculations; they included single and double excitations from a multireference list of configurations. In general, the RCI included all lowlying X-s states of the same s2 symmetry as reference configurations. For example, the RCI calculations of a state of 0; symmetry could include as reference configurations those configurations that describe lZ:(O;), 311,(0+),3Z-(0,+),etc. and other states of 0: symmetry. The RCI taus differs from a normal CI in that in the ordinary CI, 311 cannot mix with the l2: or 32;states. Most of the tkeoretical calculations made after 1985 were done by using the complete active space MCSCF followed by multireference singles + doubles (CASSCF/MRSDCI/RCI) methods or the first-order CI method employing valence Gaussian basis sets of higher than double-{ + polarization quality. Recent calculations on Inz,Sb2,12,etc. were done with relatively larger basis sets compared to calculations on Ga2, Gez, As2, and Se2. In the CASSCF method, a set of the most important electrons for chemical bonding (active electrons) are distributed in all possible ways among orbitals referred to as the internal or active orbitals. The active orbitals are normally chosen as the set of orbitals that correlate into valence atomic orbitals at infinite separation of the various atoms in the molecule. The CASSCF method thus provides a zeroth-order starting set of orbitals for inclusion of higher order correlation effects. The higher order electron correlation effects not included in the CASSCF are taken into account by using the configuration interaction method. The CI calculations are carried out by using the second-order CI method (SOCI), the multireference singles + doubles CI method (MRSDCI), and the first-order CI method (FOCI). The first two methods are more accurate in comparison to the last method. The SOCI calculations included (i) all configurations in the CASSCF, (ii) configurations generated by distributing n - 1 electrons in the internal space and 1electron in the external space ( n = number of active electrons), and (iii) configurations generated by distributing n - 2 electrons in the internal space and 2 electrons in the external space in all possible ways. The MRSDCI calculations included a subset of configurations determined by the important configurations in the CASSCF (coefficient 2 0.07) as reference configurations. Then single + double excitations were allowed in the MRSDCI. A less accurate method compared to MRSDCI labeled POLCI includes all configurations in the MRSDCI except those generated by distributing 2 electrons in the external space. We use acronyms such as CASSCF, SOCI, FOCI, MRSDCI, SCF/RCI, POLCI, etc. in this article to describe the nature of the calculations done on various species. Readers are referred to this section and to other review^^^?^^ to comprehend the meanings of these acronyms.
96 Chemical Reviews, 1990, Vol. 90,No. 1
I I I . Spectroscopic Properties and Potential Energy Curves of Heavy Homonuclear Dimers (Ga, to Biz) A. Ga,
TABLE 1. A Few Low-Lying Configurations and the Electronic States Arising from Them for Gal YO c o n F i g u r a t i o n
2 2 io g l a u 2 0 g i n d
Early investigation of Ga2 was made by Ginter et a1.,lz2who obtained absorption spectra in a furnace in the 19000-21 000-cm-' region. Later, Douglas et al.123 reported electronic absorption spectra of Alp,Gaz, and Inz. Two absorption bands, one in the range 1300015600 cm-' and the other in the range 24 000-30 000 cm-l, were observed. These authorslZ3concluded that the ground state of these dimers was lZ;. Ab initio calculations on AlJZ4have revealed, however, that the 311ustate is the lowest state and lZ; is much higher in energy. Basch et al.lZ4reassigned the two electronic bands of Ai2 observed by Douglas et al.lZ3to this 311g 3Puand 311 (2) 311utransitions. The vibrational frequency of the ground state of Ga2 was reported by Froben et al.lg2 Related B2127J29 and Alz124J26J30 molecules have also been studied. Thermodynamic studies of Gaz and related molecules131J3zrevealed that the dissociation energy of Gaz is about 1.4 eV. Since the electronic states of Ga, have not been well characterized, earlier thermodynamic calculations of De assumed an incorrect ground state and partition f ~ n c t i 0 n . lThe ~ ~ Gaz dimer has also been observed by laser irradiation of a GaAs crystal125and in a supersonic jet beam.97 Bala~ubramanian'~~ carried out CASSCF/FOCI calculations on Gaz that employed RECPs. The RECPs included the outermost 4s24p1 shells in the valence space replacing the rest of the core electrons by relativistic effective core potentials. A (3s3pld) valence Gaussian basis set was employed for the Ga atom. The d exponent was optimized for the ground state of Ga2 at the CASSCF level. More recently, Balasubramanian402studied many electronic states of Gaz+and Ga;. In the earlier as noted in an erratum, there was an error in one of the ECP parameters which led to longer bond lengths. However, this was subsequently corrected. Table 1 shows the possible low-lying states of Gaz while Table 2 depicts their dissociation limits and atomic energy separations" at the dissociation limits. As seen from these tables, there are a t least 20 low-lying states of Ga2 in the absence of the spin-orbit term. Table 3 shows the corrected spectroscopic properties of bound states of Ga2obtained by Balas~bramanian.'~~ Among the 18 electronic states investigated by Balasubramanian, the 3Au, 32i,3Zi(II),311g,and lIIgcurves were found to be repulsive. The calculated potential energy curves of some of many states are shown in Figure 1. As seen from Table 3, the calculated ground state of Ga2is 311u. However, the T, value of 32, is only 410 cm-'. The lZ+ state is 4734 cm-l above the 311u state. Evidently, tkis states does not appear to be the ground state of Gaz. Consequently, Douglas et al.'slZ3 assignment of lZ; to the ground state of Ga2 (Al,) is inconsistent with Balasubramanian's electronic structure calculations. Table 4 shows the adiabatic and vertical transition energies of allowed electronic dipole transitions for Gaz. For Gaz, Douglas et al.lZ3reported two absorption bands, one in the 15590-cm-' region and the other in
-
Balasubramanian
-
u
o
1s 2 lo 2 !n I n g
u
u
3
l o 2 l o 2 a in 2 g
"
g
U
l a 2 l o 220 2 g
u
"
g
9
9
b
.
'Au,
, 'r:
'E;
3 A u , 'E;,
'LU, ' Z J
3nn, l n Y
9
1
1
zg, A g $ 2 ;
1 -
2
zE
3;. 12 ;
20 9
'E;,
5 E -L , 3 Z-u ( 2 ) . 'E;,
3 -
:a l o 2 0 9 U L
1,*2!o220
,:E'
y
2 2 2 l a 10 i n 9 u 3 2
3Au,
9
l o 2.1 g 2U 2 n2d g
2
liiu
1-t
g
1oZlaZ2a I n g
U'
3 - 1 1 E g > A g > 2;
2 2 2 la lo in s
1-8 states
"
TABLE 2. Dissociation Relationships for a Few Low-Lying States of Gal 1-8
states
dissociation l i m i t atomic l e v e l s energi Expt
nFrom ref 134.
-
-
the 29 934-cm-' region. These bands were assigned by those authors to AlII, XIZi and BIZi X'Z;, respectively. However, as seen from Table 4,the 'II, state is below the lZ; state. Further, the most probable candidate for the ground state is 311u. BalasubramaniarP3 argued that the most consistent allowed electric dipole transition for the 15590-cm-' is the A31& X311, transition. The absorption bands in the 29 000-cm-l region could be assigned to the 311,(11) X311utransition. The theoretical Tvert(29 102 cm-l) is most consistent with the observed bands in this region. Ginter et a1.lz2 reported emission bands in the 19500-22000-cm-' region for Gaz. It can be inferred from Tables 3 and 4 that the most probable transition for these bands is the 3X;(I) 3X,(I) transition. The theoretical transition energy for this transition (21 191 cm-') was found to be in very good agreement with the observed bands in this region (19 000-21 000 cm-l). The theoretical dissociation energy of the ground state of Gaz obtained by Balasubramanian133was found
-
-
-
Chemical Reviews, 1990, Vol. 90, No. 1 97
Heavy p-Block Dimers and Trimers
-
TABLE 3. FOCI SDectroscoDic ProDerties of Ga2
0.06-
-
0.05
2.752
0
I58
1.18
2.506
446
197
1.13
2.802
4037
145
0.68
2.954
4129
129
0.67
2.584
4274
168
0.65
0.04-
-
-
0.03
; In
0.02-
c L
-
0.01
0
I
*P+*P
T w
O-
2.912
8267
125
2.681
20 978
166
2.702
2a 772
210
-0 03
3.495
29 877
86
-00 4
2.407
30 325
292
-005
2.757
35 705
210
3.261
36 722
58
2.471
37 865
315
2.773
40 125
164
3.063
43 383
84
2.389
45 318
I53
2.83i
45 a75
125
3.414
44 587
84
2.7C3
54 C52
353
-
-
3zi
I
I
3.00
I
4.00
I
500
6.00
R
Figure 1. Potential energy curves for the electronic states of Gaz (reprinted from ref 133; copyright 1986 American Chemical Society). See Table 3 for spectroscopic labels of experimentally known states. TABLE 4. Vertical and Adiabatic Excitation Energies for Dipole-Allowed Transitions" Electronic Transitions
-
~~~~
to be 1.2 eV. This was in reasonable agreement with the thermodynamic value of 1.4 eV.1313132However, the thermodynamic value needs to be recalculated by using the correct partition function for Ga2. Balasubramanian's calculations on Ga2 should be useful in calculating the partition function. A number of low-lying allowed electric dipole transitions exist for Ga2, many of which have not yet been observed (Table 4). Specifically, 32g 3Z;(II) is an allowed dipole transition, which falls in the region of 41 000 cm-'. The theoretical transition energy should be slightly higher than the true energy. The 32-(11) X3HUand 32JI) X311, transitions are allowed in the perpendicular direction. The former transition energy is in the 28 000-cm-' region. Transitions originating from the lIIUelectronic state to the lIIgas well as lII,(II) states should also be investigated. Table 5 depicts the contribution of various electronic configurations to the electronic states of Ga2. As seen from Table 5, the ground state (311u)as well as lII, is dominantly l a $ ~ ~ 2 a ~ l r ,The . 3Z; and 'Ag states arise from the l a i l a i l r t configuration. The l2: state exhibits an interesting behavior as a function of internuclear distance. At short and near-equilibrium distances this state is a mixture of l a 2 1 0 % land r ~ la21a2,2aiconfigurations. Consequently, t i e R, value of the 12; state is quite different from the Re values of the 32-and lAg states, which are predominantly l a ~ l a ~ lAt r ~longer . distances, lZ'(11) exhibits approximately the opposite behavior in tkat l a ~ l a ~makes l r ~ a significant contribution at long distances.
-
-0.01
2.00
" From ref 133. The A state observed in the A X broad band was system assigned to the 3 ~ m %,, t r a n ~ i t i 0 n . I ~ ~
-
-
-
T,
(cm-1)
~
A311
+
9
3r;
X3nU
14 147
b
22 052
21 191
41 570
41 307
1 288
410
3Z;g ( I I )
32 419
27 954
3
29 102
27 565
3r;
3nu
B3n (11) * x nu 9 In" I,+ 9 Inu * Illg
1 029
1 026
12 248
D
Illu * lng(II)
31 172
29 493
t
" From ref
133. Final electronic state is not bound.
The Rydberg configurations make substantial contributions to the 3Zi(III), 32i(II), and 3Au states a t short distance. For the first two states, the contributions from Rydberg configurations are 23% and 16%, respectively. The theoretical T,value for the 3Zi(III) state should therefore not be regarded as accurate since the valence basis set needs to be extended further for proper description of Rydberg orbitals. For the 32; and 3Z;(II) states of spectroscopic interest, the contribution of the Rydberg configurations a t the equilibrium geometry is small. The total contribution of Rydberg configurations for the 3Z;(II) state at 2.50 A is 4%. The total contribution of Rydberg states for the 32; state a t 3.00 A (equilibrium geometry) is 9%. At long distances, however, these states eventually become Rydberg states, as their dissociation limits include Rydberg atoms. More recently, Shim and Gingerichm completed all-electron calculations on Gaz and Ga3. Their results are in good agreement with Balasubramanian's results on Ga2 shown in Table 3.
98 Chemical Reviews, 1990, Vol. 90,No. 1
Balasubramanian
TABLE 5. Contributions (70) of Various Configurations to the Electronic States of Gaz
31-
9
Inu
Id
9
3.00
00
5.00
10
1.57
22
2.10
U8
2
4.00
12
19.3
3.00
08
5.00
6
2.5
27
t i
2.85
85
3.00
51
3,011
411
32
1.25
Jn
38
3.15
'12
18
i.no
51
3.15
A' u 311
g
5.6 37
4.2 4.2 15 38
71
40
31
32.5
15
48
8.3
12 10
2.80 3.00
R
91
5 . R0
2 . I0
60
i.w
GO
4.00
il 4
60
8.6 I2
1
2
3.uu
18
64
5.00
14
58
3.00 5.00
54
5
56
29
4.00
73.3
31
13.7
66
I1
7.50
4
2.10
3
74
3.00
2
70
76
3.00
41.5
16
5.00
50
29
2.55
90
2.131)
91
16
2
2.25
79
7.55
43
21
2.110
3
7.5'J
33
49 24
7 . flu
69
7.4
2 DO
76
4
6. Ge,
The only spectroscopic study of Ge2appears to be the Raman and fluorescence spectroscopic investigation of Froben and S ~ h u 1 z e . lThe ~ ~ other experimental study on Ge, is that of Gingerich and c o - ~ o r k e r s which '~~ deals with the dissociation energy of Ge,. Shim et as well as Pacchioni13*have studied the lowest lying states of Ge, using the Hartree-Fock (HF)/CI method. The molecular orbitals for CI calculations were obtained by a single-configuration SCF treatment. As Shim et al.13' noted, starting with MOs obtained by different states led to different results in this method. Thus, a single-configuration SCF treatment does not seem to be very reliable for obtaining the spectroscopic properties of excited states. Also, the splitting between the 32- and 311ustates is small for Ge,, analogous to Ga,. b a l a ~ u b r a m a n i a nmade l ~ ~ CASSCF/FOCI calculations on Ge, employing RECPs that retained the 4s24p2 outer shells in the valence space. A (3s3pld) valence Gaussian basis set of the same quality employed for Ga, was used for Ge,. Table 6 shows a few low-lying MO
3.3
1
configurations and a list of possible X-s states for Ge,. Table 7 shows the dissociation limits for the low-lying states of Ge,. As seen from these tables, there are many low-lying states for Ge,. Table 8 shows the theoretical spectroscopic properties (Re,T,,u e )of 14 low-lying electronic states of Ge, obtained by Ba1a~ubramanian.l~~ Figure 2 shows the potential energy curves of some of these states. As one can see from Table 8, the ground state of Ge, is X32;; however the 311, state is only 767 cm-' above the 32:9 state. Since this splitting is small, the possibility of 311u being the ground state cannot be ruled out since higher order correlation corrections not included in Balasubramanian's calculations139may reverse the ordering. Table 9 shows the adiabatic transition energies of some allowed electric dipole transitions. The most important transition originating from the ground state (32;)is the 32; X3Z; transition. The theoretical energy separation for this transition is 20 979 cm-l. The corresponding transition was observed for Sn, at 18222 cm-l.140 It seems that to date this transition has not been observed for Gez. The spectroscopic bands in this
-
Chemical Reviews, 1990, Vol. 90, No. 1 99
Heavy p-Block Dimers and Trimers
TABLE 6. A Few Low-Lying Electronic Configurations of Ge2 and the A-s States Arising from Them'
TABLE 8. Theoretical Spectroscopic Constants of Geza State
MO c o n f i g u r a t i o n
(A)
R,
T, ("1)
w e (cm-1)
A-s s t a t e s
3 5
2.44
0
259
3% 1 A9 1
2.34
767
251
2.50
3982
255
2.44
5521
192
1
2.35
5794
269
I + Zg(W
2.41
10 840
258
3
2.88
13 534
152
3n 9
2.67
13 995
157
37g ( I I )
2.69
17 373
165
1
2.95
17 455
99
3r;
2.93
20 979
I52
3 ~ ; (1 1 ) 1 A"
3.68
24 634
100
2.86
29 078
156
hJII)
3.25
32 563
125
zg'
2a g2 l a u l a g
A' ,
32;,
3El,
'E:,
'Au,
%
'1;
"The la$~i shell is not shown. TABLE 7. Dissociation Relationships for a Few Molecular States of Gel ~
Atomic States
Molecular S t a t e s
~~~
Energy o f t?e separated atomsa
"The experimental D: = 2.65 eV (Kingcade et al.136)compared to CASSCF/FOCI D,(Ge,) = 2.3 eV. All spectroscopic constants are from ref 139. , .u,
3P
+
3P
0.0
i
33iC
3P +
3
",(3),
7125
3 3 A g ( 2 ) , A u ( 2 ) , 3+g, 3 ~ u 3330
"From ref 134. OClO
-
region should be intense and well resolved. The yIIg 311utransition is also important. Balasubramanian's calculations predicted this transition to be in the 13000-cm-' region. The 311u 32;transition energy for Gez is much lower than the corresponding 0: 0: transition for Sn2 or Pb2. The l2' state may be observable with small probalZi(0') emission, since Gez is a bility in t8e 3Z;(O:) reasonably heavy molecule. "he theoretical energy for this transition is 15460 ~111-l.'~~ If the 32; state can be populated, then this transition should be observable in lZ'(I1) emission. In the same way, the 3Z;(O:) transition should be observable in the 10 140-cm-el region. The 311,(11) 311utransition is also an important allowed electric dipole transition. As seen from Table 9 this transition occurs at 16503 cm-l. The electronic spectra that correspond to the lIIg 'nuand lIIU(II) lIIg transitions should be in the regions of 12 204 and 15 198 cm-l, respectively. The theoretical CASSCF/FOC1139De value for the ground state (X3ZJ of Ge2 is 2.29 eV. This is in reasonable agreement with the experimental value of 2.65 eV reported by Gingerich and co-workers.136 The small discrepancy between the calculated and observed values
-
-
-
-
-
-
-
1
W
3c nOI0
-0330
050
09C 2C
3C
40
50
60
70
R-(A)
Figure 2. Potential energy curves for the low-lying electronic states of Gez (reprinted from ref 139; copyright 1987 Academic Press, Inc.).
could be attributed to higher order correlation corrections as well as the use of effective core potentials. Balasubramanian's D239was found to be close to the HF/CI calculation of P a ~ c h i o n i ,which ' ~ ~ yielded a De value of 2.34 eV. Balasubramanian's vibrational frequencies of the 32; and 311ustates were in reasonable agreement with those of Shim et al.'37 The value reported by P a ~ c h i o n i 'for ~ ~ the 311u state (195 cm-'), however, is different from Balasubramanian's and that of Shim et al.13' For both Snz and Pb2 the vibrational frequencies of 32; and 311u differ by at most 10%.
100 Chemical Reviews, 1990, Vol. 90, No. 1
Balasubramanian
TABLE 9. Adiabatic Transition Energies of Allowed Electric Dipole Transitions of Ge,n Transition
Energy
(ern-')
Transition
Energy (cm-')
TABLE 11. Dissociation Relationships for a Few Low-Lying States of As2 Molecular S t a t e s l,+ - g ' 3 I;, 5 E;,
Atomic S t a t e s 4s
7 1;
+
4s
Energy (cm-'ia
0.0
32-g ' 3 T -U ' 3" g ' 37" j",
"z;,
5ng, 5 i7"
TABLE 10. Contributions of Various Configurations to the Electronic States of Gez at Their Equilibrium Geometries" State
2 2 2 0 In (85%), 2021n2 ( 2 4 ) g :
g
g
"
3,-
lation seems to be a bit more important for the 311ustate than the 329 state. Consequently, higher order correlation effects may lower the 311ustate a bit more compared to the 3Z- state. The 32-state is a mixture of the 2u~1xu17rg, 2agbauln~, and laulag Y configurations, suggesting the existence of another 32; state that would also be a mixture of these and other configurations. Although Bala~ubramanianl~~ did not calculate the energies of this state at all distances, on the basis of the splitting of this state at 2.75 A from the equilibrium geometry of the ground state it was predicted that the T , value of this state should be below 29000 cm-'.
c.
Lu
3:
Experimental atomic energy separations from ref 134.
Configurations
(
:I)
-7,::1
nFrom ref 139.
However, Pacchioni's we for 311 is 17% smaller. Table 10 shows the contributions of various MO configurations to the FOCI wave function of the electronic states of Gel. As seen from Table 10, the 32- and lAg states are predominantly 2ailxt. The 311uana 'nu states are composed predominantly of the 2a la: configuration. Like Ga2,the '2; state of Ge, exkibits an interesting behavior as a function of internuclear distance. At short distances, it was found to be predominantly made of the la: configuration. A t near-equilibrium geometries this state was found to be a mixture of 2 a ~ l 7 and r ~ la: configurations. The lZi(I1) state is also a mixture of these two configurations, with the latter configuration making a greater contribution at its near-equilibrium geometry. Thus, these states exhibit avoided crossings. As seen from Table 10, corre-
As,
Electronic spectra of As2 were first observed by Gibson and MacFarlane7' in 1934. These authors7' studied the absorption spectrum of As, in the 40 500cm-l region. The vibrational levels of the excited state participating in these bands were highly perturbed and predissociated, which led to the prediction of crossing of another potential energy curve with this curve. Later, a number of authors studied the As, molecule experimentally.141-154J58These studies included both absorption and emission spectra of As2 in the visible and vacuum-UV regions. Particular systems that have been observed include A X, C X, C A, a X, B X, d X, and D c. Among these, the A X system has been extensively studied. Vibrational and rotational analyses of a number of observed systems have been carried O U ~ . ~Heimbrook ~ ~ J ~ et al.l& studied the Asz dimer generated by evaporating gallium arsenide which was then trapped in solid neon. These authors156 studied vibronic bands of three systems of As2 designated as the c X, e X, and a X systems. Both the ionization p~tential'~'and the electron affinity148 of As2 have been measured. Theoretical calculations on As2 include those of Kok and Hall,'% Watanabe et al.,lS7and Balas~bramanian.'~~ The first two works were restricted only to the ground state of As2.155*157Bala~ubramanian'~~ carried out CASSCF/FOCI calculations on 18 electronic states of As2 using RECPs (4s24p3shell of As included explicitly
-
-- - - -- -
- -
-
Heavy p-Block Dimers and Trimers
Chemical Reviews, 1990, Vol. 90, No. 1 101
TABLE 12. Spectroscopic Properties of As$' Re ( A )
State
Theory
Expt.
Te (cm-') Theory Expt.
0.140-
we (cm-')
Theory
0.120-
Expt. 0.100-
0
394
0.080-
11 860
14 500
235
0.080-
2.357
19 976
19 915
324
2.58
25 053
--
315
2.345
26 406
24 641
341
2.756
30 676
--
179
-0.020
2.313
30 743
--
341
-0.040-
2.561
32 722
--
274
-0.060-
2.298
33 262
--
317
-0.080-
2.654
41 228
--
221
2.466
43 297
[40 3491
253
R&
2.561
44 669
--
290
2.535
45 327
--
?;6
2.747
$7 668
-_
Figure 3. Potential energy curves for the low-lying singlet electronic states of As2 (reprinted from ref 159; copyright 1987 Academic Press, Inc.). See Table 12 for spectroscopic labels of known states.
174
2.74
51 294
--
198
2.164
0
2.418
jt '
0.040
I
' E p
-
'A"
0 020-
w 0.0004s * 4s
-
-0.100, , 1.52.0
, , 3.0 4.0
,
,
5.0
6.0
-
, 7.0
,
I
I
8.0
0.190
a Theoretical spectroscopic constants are from ref 159. Experimental values are from ref 37.
0.130-
in valence) and a valence (3s3pld) Gaussian basis set. 0.110The CASSCF active space included four al, two b2, and two bl orbitals in the Czogroup. The FOCI calculations 0.090included up to 27 OOO configurations while the CASSCF I calculations included up to 384 configurations. 0.070Table 11shows the possible low-lying electronic states 0.050of Asz and their dissociation limits. As seen in Table 0.03011,due to the 4s24p3open-shell ground state of the As atom, As2 has the largest number of possible valence electronic states of all the dimers in that row. The 4s 4s ground state of As2 is a '2: state arising from lailc~~2c~~17rt, which represents a triple bond between -0.0301 As atoms. =; -0.0504 . Table 1 2 shows the spectroscopic properties of a 1.5 2.0 3.0 4.0 5.0 6.0 7.0 6.0 number of low-lying states of As2. Also indicated in that table are the assignments of the states that have been R (A) observed experimentally. Figure 3 shows the potential Figure 4. Potential energy curves for the low-lying triplet states energy curves of low-lying singlet states, and Figure 4 of As2 (reprinted from ref 159; copyright 1987 Academic Press, shows the potential energy curves of the triplet states. Inc.). See Table 12 for spectroscopic labels of known states. As seen from Table 12, the theoretical bond length of the ground state is -0.06 A longer than the experila21a~2ag2uul?r~, and la~la~l?ru17r3 configurations. mental value. The slightly longer bond was attributed 1 s already mentioned, the A * bands are predisby B a l a s ~ b r a m a n i a nto l ~ higher ~ order correlation efsociated and have been studied by many authors. Bafects that were not included in the FOCI calculations las~bramanian's'~~ theoretical calculations supported and the use of RECPs. the existence of a B state near the A state, which had The strongest electric dipole transition from the the same symmetry, since the A state is a mixture of ground state is the A'Z: X'Z: transition. The first several configurations. The exact nature of the state 'Z: state is calculated 43207 cm-' above the ground predissociating the A * X bands has not yet been esstate (Table 12). This has been characterized experitablished. mentally as the A * X system in the 40 350-cm-' region. Heimbrook et al.156observed vibronic bands of Asz in the red, yellow, and blue-near-UV regions. These The calculated o, and Re values of the A state are in very good agreement with the experimental values. The transitions were labeled c * X12:, e * X, and a * X. ' 2 : state is a mixture of the l a $ ~ ~ 2 a ~ 1 7 r ~ l ? r ~These , authors'56 also tentatively assigned the a, c, and
i
~
I
I
I
,
-
-
,
I
I
I
102 Chemical Reviews, 1990, Vol. 90,No. 1
Balasubramanian
TABLE 13. Leading Configurations of the Various Electronic States of As, at Their Equilibrium Geometrieso Stste
Corilguration
this transition lies at 51 300 cm-'. The theoretical W , value of the 'II,(II) state is 108 cm-l, indicating the broad nature of the potential surface of the 'II,(II) state. A number of unclassified bands have been observed158in the 47 000-55 OOO-cm-' region. Thus, the absorption bands in this region could tentatively be assigned to the lII,(II) X'Z; transition. There are many allowed electronic transitions originating from excited electronic states that appear not to have been observed. The 311u 311gtransition is a strong allowed transition and could be observed as the 311u 311gemission system. The approximate energy for this transition is 18616 cm-'. The 3A 3Au transition occurs at 10728 cm-l, and the lA,, 64transition could be observed in the region of 17 000 cm-l. None of the bound excited '2: states (Table 12) have been characterized experimentally. Emission from the A'Z; state to all lower '2' states is allowed. The A'Z: 'S'(I1) emission coufd be observed at 10485 cm-'. The 'Z!(III) and 'Zl(11) states would be accessible in emission from the BISistate. The +IIg lIIUand 'II lII,(II) transitions could be observed at 12065 and 18032 cm-', respectively. None of these transitions have been observed. The theoretical De of the ground state of As2obtained by using the CASSCF/FOCI method159was found to be 21 860 cm-' or 2.71 eV. The spectroscopic De value was obtained by extrapolating the predissociated A-X bands into the 4S+ 2Datomic ~ t a t e s . ' ~ ~The 3 ' ~ De value obtained this way (3.96 eV)142should not be regarded as accurate, and usually such extrapolations provide only an upper limit. Nevertheless, second-order correlation corrections could increase the De value of the ground state. The De of As2 should, however, be closer to 3.8 eV, and thus CASSCF/FOCI calculations underestimate the De of As2. Table 13 shows the important configurations contributing to the electronic states near the equilibrium geometries of the various states.159As seen from Table 13, the ground state of As2 was found to be predominantly the triple-bond configuration la~la:2a~17r~, although the l a ~ l a ~ 2 u 2 1 aconfigurations ~la~ also make an appreciable contribution, thereby lowering the bond order to 2.64. In Table 13 the sums of the squares of the various configuration spin functions compatible with the electronic distribution are shown. The A'Z; state participating in the strong A X system is a mixture of 1 u ~ 1 u ~ 2 a 2 1 a, ~l ul a$ ~ ~ 2 a ~ 2 a , l and a ~ , other configurations. %he cfZi state is predominantly lu21a22a217r317rg.The a3Z; state is composed of the lagl~,2ag17r,17rg B Y B Y configuration. From the contributions of the A'Z: state, it can be deduced that the orthogonal BIZ: state should be a mixture of the la~la~2ag2au17r~, l a ~ l a ~ 2 a ~ l n , land 7 r ~ la~la~2a217r~17rg , configurations; the first configuration would prosably make the highest contribution to the B state.
-
-
-
--
-
-
-
.Zac' ,
N u m b e r s in p a r e n t h e s e s a r e c o n t r i b u t i o n s in p e r c e n t a g e ; f r o m
ref 159.
e states to the 'Z;, 3Zi, and 3A, states. As seen from Table 12, theoretical calculations support the assignment of the a, c, and e states to the 32;, 3Z:, and 3Au states. The calculated W, values of these states are in reasonable agreement with the experimental results. The b X system was observed by Almy and cow o r k e r ~ ,although ~ ~ ~ - ~no ~ spectroscopicproperties have been reported for the b state, since the b X system appears to be weak. A possible candidate for the b state '~~ is a 311ustate. Balasubramanian's c a l c ~ l a t i o n s supported the existence of a 311ustate whose T,value is 44669 cm-' and was designated as 311u(II). The experimentally observed bands for the b X systems are consistent with the theoretical T, value. Although the 311u lZ; transition is not allowed in the absence of spin-orbit interaction, the spin-orbit effects seem to be large enough to give rise to the weak 0: lZi transition, where 0: arises from 311u. Alternatively, the 311,(1,) '2; transition should also have been observable in the perpendicular direction via spin-orbit mixing of 311uwith Ill,. Many electronic states are reported in Table 12 that have not yet been observed. Specifically,the 'II, l2: transition is an allowed electric dipole transition. The calculated Tevalue of the 'nustate is 45 327 cm-'. Thus, this system should be observable in both absorption and emission. Sibai et al.I5O have observed bands in the 42 400-44 500-cm-I region. The theoretical Te value of the 'II, state seems to support the assignment of these bands to the 'II, X transition, although this assignment is not yet confirmed. The lII,(II) X I Z i transition is also an allowed electric dipole transition. From the calculated properties of the lIIu(II)state (Table 12).
-
-
-
-
-
-
-
-
-
-
D. Se,
The Se2dimer has been studied by a number of inv e s t i g a t o r ~ . 'The ~ ~ ~earlier ~ works on the spectroscopy of Se2 are summarized in ref 37. The O2 analogues of the Schumann-Runge bands that are attributed to the B-X system are observed for Se2 in the 26000-cm-' region. These bands for Se2were found to be perturbed by an A states1@The B-X, A-X, and a-x systems have
Heavy p-Block Dimers and Trimers
Chemical Reviews, 1990, Vol. 90, NO. 1 103
TABLE 14. A Few Low-Lying Electronic Configurations of Sez and the X-s and w-o States Arising from Them"
TABLE 15. Dissociation Relationships for a Few Low-Lying States of Sez ~
Electronic configuration 20
2In4i n2 9
U
A-s states 3,-
3
Q
states
W-w
o+ 1 9'
Atomic states
Molecular states
9
0.0
1
29
*g
20;!n31n;
3 *U
3f U
3,
of 1he atoms (cm-
Energy
3,,
zu, lu
0;- 1"
+
3P+b
&U
9576
1
A"
"From ref 134. ~~
TABLE 16. Spectroscopic Properties of Seza
i, U
I
R,
(A)
Calcd.
x 32; 0; x1
2.244
2.166
0.0
357
2.240
---
664
360
4652
341
8557
323
---
15241
246
15651
244
(2.71)
2.51
21277
197
(2.72)
2.54
21961
194
2.27 2.30 2.51
;
: 0
"The 1o:la; shell arising from the valence s is not shown.
we (cIR-~)
Expt.
lg x2
2U
(cm-')
Calcd.
'2;
IA,
T,
State
2.52
Expt.
Cilcd. Expt.
22837 (2.73) -also been observed in a matrix by Bondybey and Eng232 24822 1 i ~ h . lProsser ~~ et al.164have observed a BO: -b1Z; 2.54 2.44 system by populating the B and A electronic states with 133 30567 3.12 -a 4131-A krypton ion laser. In addition, a C-X system 122 31582 3.01 -was also observed in the 53000-cm-' region. Below 26000 cm-', five A-s states of Se2 and some of their -33325 3.32 -spin-orbit components have been observed. These -33712 3.40 -states were given the labels B, A, a, b, and X, among which B, A, b, and X have been tentatively assigned. 233 36814 2.53 -The nature of the state perturbing the B3Z; state is 39173 325 2.58 -not well-known. This has been the subject of many i n v e s t i g a t i ~ n s . ' ~ ~Further, J ~ ~ J ~ ~unlike 02,the nature 1$1 41270 2.02 -of the electronic states below the B state is not clearly 42556 159 2.90 -understood. Theoretical study of the low-lying electronic states of Se2was made by Balasubramanian with "Experimental values for the XI, X , and B states quoted here the objective of comprehending the nature of the loware from Huber and Herzberg's Experimental values for lying electronic states of Sez.172He assigned the exthe lAg, lZ:(O;), 311,(0:), and 311,(1,) states are from ref 168, 164, 165, and 162, respectively. Theoretical values are from ref 172. perimental spectra by a comparison of calculated and observed values. Saxon and L ~ u have ' ~ ~shown that such theoretical calculations yield reasonable spectroploying (3s3pld) valence Gaussian basis sets and scopic constants for O2 even if errors in De may be RECPs that included the outer 4s24p4shells explicitly appreciable. Experimental study of other related group in the valence space. The internal space consisted of VI and group IV dimers is also quite i n t e n ~ e . l ~ ~ - l ~ ~four a1 orbitals, two b2 orbitals, and two bl orbitals for Bala~ubramanian'~~ carried out CASSCF/FOCI calSe2. The FOCI calculations of Se2included up to 10750 culations on 20 low-lying electronic states of Se2 emconfigurations. 139
104 Chemical Reviews, 1990, Vol. 90, No. 1
Balasubramanian
arising from the l a ~ l a ~ 2 a ~ l a configuration. ~17r~ The discrepancies in theoretical Re and we values of the ground state are within 3-6% of the experimental values. The w,s of the various states in Table 16 were calculated by using a cubic polynomial fit of energies at near-equilibrium geometries. The small discrepancy between calculated and observed values arises partly from the errors in fits (usually within k15 cm-') and partly from higher order correlation corrections and basis set effects that influence the we values. The strongest electronic transition originating from the ground state is the B3X; -X3Z; system. The theoretical properties of the 32; state are in very good agreement with the spectroscopic values, confirming the earlier assignment of this system. There have been a few experimental investigations on electronic states below the B state. Specifically, the A state in the A-X system was identified by Heaven et al.169The T, value of the A state reported by these authors is about 24 111 cm-'. Bala~ubramanian's'~~ theoretical calculations confirmed the existence of a 311,(0:) state with a theoretical T, of 21 277 cm-' in this region. Thus, the A state The was found to be most consistent with 311,(O:). Figure 5. Potential energy curves for the low-lying states of Sez (reprinted from ref 172; copyright 1987 American Chemical So311,(0:) spin-orbit state could mix with 3X;(O:), alciety). See Table 16 for spectroscopic labels of known states. though Balasubramanian's RCI calculations revealed that this contamination is negligible for 311u(O:) at its Table 14 depicts the possible low-lying electronic equilibrium geometry. Similarly, the mixing of the states of Se2 in both A-s and w-w coupling. Table 15 311u(lu), 3X;(lu), and 'IIU(lu) state was found to be relshows the dissociation relationships for the low-lying atively small for the selenium dimer. Balasubramanielectronic states of Se2. The ground state of Se2 is an's172 theoretical Re value for the A state did not agree evidently an X3Xg state arising from the well with the value calculated from the experimental 10$3ail7r37ri configuration analogous to 02.Many spectroscopic data. A similar deficiency was found in of the low-lying electronic states enumerated in these the 3JIustates of Br, and IB. The 311ustate is weakly tables were studied by Balasubramanian. bound and its surface is shallow. The basis set that was Spin-orbit interaction was also introduced by Balaused in conjunction with effective core potentials does ~ u b r a m a n i a n to ' ~ ~some of the electronic states of Se2 not appear to be of sufficient quality to compute the (32;,'El, 311,(0:), 311u(lu)) with the objective of estispectroscopic properties of weakly bound excited states. mating the effect of spin-orbit interaction. This was Diffuse basis functions and polarization functions could accomplished through the RCI scheme. A double-l be important for the 311uand 'nuexcited states. Bondybey and Eng1ishlB have observed spectroscopic STO basis set optimized for the 3Pground state of the bands with a T , value of 16321 cm-', which they atSe atom was employed in SCF/RCI calculations. The tributed to a weak electronic transition. These authors w-w states arising from the 3E;, 'Ag, and '2' states were obtained by using the orbitals generatea from SCF call this an a-X system, although this designation could also be used for the 4000-cm-' system. To differentiate calculations of the 3E; state. The w-w states arising from the 311uand 'nu states were calculated by using the two systems, the state with T, = 4000 cm-' was denoted by Balasubramanian as a'. Theoretical calcuthe 311uSCF orbitals since 32;SCF did not produce a lations suggested the existence of a 3Au state in the good 20, orbital as this is unoccupied in the 3E; state. neighborhood of the observed electronic bands. The The RCI calculations of the 0; state of Se2included theoretical and experimental we values were also found single and double excitations from Cartesian reference to be in very good agreement. Further, the 3Au 3 X ; '2;) configurations arising from 10;iO~2a;17r~17r~(3~transition is dipole forbidden, supporting the experiand l 0 ~ l 0 ~ 2 0 ~ l 7 r ~ l 7 r ~ (The ~ I I ,1, ) . state include$ five mental observation of weak electronic bands in this reference configurations arising from 32;(1,), 311g(Ig), system. The above reasonings led Balasubramanian to and lIIg(14). The 2, state included eight reference suggest the assignment of this system to the a3A,-X32; configurations arising from lAg and 311F. Balasubraforbidden transition. m a n i a r ~ also ' ~ ~ carried out RCI calculations of the O,: Prosser et a1.'@ and Winter et characterized an l,, and 2, components of the 311ustate. The 0 : state electronic state (b) with a T, value of 7958 cm-'. The included limited single and double excitations from 24 b state was tentatively assigned to the '2' electronic reference configurations arising from a number of 311u state. As seen from Table 16, the results of theoretical references and 9;references. The 1, state calculations calculations support this assignment. The fact that the included 311u, Ill,, and 3E: references, while the 2, theoretical and the experimental T, values of this state calculations included only 311ureferences. are in excellent agreement and that there is no other Table 16 shows theoretical and experimental specelectronic state with this T , value led Balasubramanitroscopic properties (Re,T,, and w e ) of the low-lying an172to confirm unambiguous assignment of the b state electronic states of Se2. The potential energy curves of to the lX; state arising from the l a i l 0 ~ 2 0 ~ 1 7 rcon~li~~ some of these states are shown in Figure 5. As seen figuration. from Table 16, the ground state (X) is the 3Z; state i2
.,A
-
Chemical Reviews, 1990, Vol. 90, No. 1 105
Heavy p-Block Dimers and Trimers
Yee and Barrow'70 studied the absorption and fluorescence spectra of gaseous Sez in the 5145-4880-A region. These authors assigned these bands to an n a transition where the upper state is believed by these authors170to be a relatively weakly bound 1, state which perturbs the B3Z;(O:) state. The a state is predicted to be approximately 4000 cm-' above the X3Z,(O:) ground state and was tentatively assigned to 'A, based on the lower vibrational frequency of this state compared to that of the ground state we. The theoretical spectroscopic properties for the 'A,(2 ) state (Table 16) certainly support this assignment. "he experimental we value (320 cm-l) does not follow the theoretical trend in Table 16, which implies that the we value of the lAg(2J state should be 20 cm-' less than the we value of the 3Z;(O+ 1 ) states. While the spin-orbit coupling of the 'A, state by 95 cm-l, it does lowers the not change the Re and we values at all. This is primarily a consequence of the fact that the 'Ag state was not found to be contaminated with other A-s states that give rise to a 2, w-w state such as 311r While it is possible that small errors could be introduced by the procedure used by Balas~bramanianl~~ to calculate wes by fitting energies using a cubic polynomial in the near vicinity of the equilibrium geometry, this was ruled out by Balasubramanian based on the fact that identical distances were used for all four states, namely, Oi, l,, 2,, and Ol(I1). Consequently, it appears that the experimental we of the a'$ state (320 cm-') is somewhat low and should be about 368 cm-'; this was arrived at by decreasing the experimental we value of the 3Z;(lg) state by 19 cm-' as implied by the theoretical calculations. The small difference in the we values of the XJ0: and X21 states is mainly due to the spin-orbit mixing of 3Z$b:) with lZ+(O,+). The 0: state was found to be a mixture of 3E; ? 8 l % ) and 'Z; (8%) A-s states in the vicinity of the equilibrium geometry. The Re of '2; was found to be larger than the corresponding value for 3Z;, and the we value is smaller than that of the 3Z; state. The Re and we values of the 3Z; and 'Zp' states when weighted with appropriate spin-orbit contaminations (81%, 8%) were found to be in accord with the theoretical Re and we values for the XIO: state in Table 16. J e n ~ u v r i e r 'has ~ ~ studied the perturbations of the bands in the B(3Z;(O:))-X system. There is some confusion introduced by using the label A to designate the state that perturbs the B state and the A3n,(O:) state. The A state that was found to perturb the B state was assigned to A3n,(l,) by J e n 0 ~ v r i e r . lIt~ is ~ best to refer to these two states with their Q quantum number in parentheses. The A311,( 1,) state is 1000 cm-' below the BO: state. There is also another state designated A' assigned to A('II,(lJ) by JenouvrieP5 which was also assigned as the state perturbing the B state. The A' state was found to be only 110 cm-' below the B state.'65 From Table 16 it can be seen that the theoretical 1, state (predominantly 311,) is 685 cm-' above the 311,(0:) state. The corresponding experimental splitting reported by J e n ~ u v r i e r is ' ~ 829 ~ cm-l and is thus in reasonable agreement with the predicted spinorbit splitting for this state. However, the larger theoretical Re value was attributed by Balasubramanian to the basis set and suggested further extension of basis sets with more diffuse functions and polarization functions since they appeared to play a significant role
-
:!' :slue
TABLE 17. Leading Configuration(s) Contributing to the Low-Lying States of Se2 at Their Equilibrium Geometries" Configuration
State
32;
za21n41n2(89%) g u 9 20 2 I n 4 In 2 ( 8 9 % ) , 20 2 I n 2 In 4 ( 6 % ) g u g 9 u 3
1
A9
2 a Z ~ n 4 i n 2( a x ) , zn2in2in4 ( 9 % ) g u a 9 u 9
I,+
3
3
2ag1nulng 2 3 3 ( g o % ) , 2 a2u l n 3u l n 3 ( 3 % ) g
AU
32; 3flu
32;
3 Zg(II)
3i79 ( 11)
37u(i~)
2 3 3 2 3 3 20 I n In ( g o % ) , 2 o u l n u l n (3%] 9 g u g 2 4 2 2 3 20 2a In I n ( 6 6 L ) , 2a 2 0 In I n ( 2 2 % ) g u u g g u u g 4 2 2a21n31r3 ( 7 3 % ) , 20 2 0 In In ( 1 4 % ) , g u 'g g u u g 2 4 2a 20 In I n ( 4 % ) g u u g 3 3 2 2 4 2a 23 In In ( 4 2 L ) , 20 I n I n (38%)), g u u g g u g 2 0 2 I n 4 I n 2 (8%) g u g 2 3 2 2 2 3 20 20 I n In ( 7 8 % ) , 20,23 In In ( 7 . 3 % ) g u u g 9 u g 2 2 3 2a 20 In In ( 4 8 % ) , 2 a 2 2 g ln41n ( 2 3 % ) , 9 u u g g u u g
especially for the weakly bound excited states. As seen from Table 16, the theoretical bond length for the B32; state is 0.1 A longer than the experimental value primarily due to the limitations of the basis set. The worst case is the 'nu state, which actually comes out to be repulsive theoretically, while the experimental A' state, which is tentatively assigned to 1rIu(lu),165 is bound, although somewhat less than the A states. More extensive calculations that include diffuse basis functions and polarization functions are needed to calculate the properties of the states perturbing the B states. Table 16 shows a number of electronic states of Sez that are yet to be observed. Specifically, the 3Z;(II) state, which has the same symmetry as the ground state, can be observed in the 32; 3Z-(II) emission system if the C state can be populated. %he 3rIu(II)state has a minimum at a long distance. The dipole-allowed 311u(II) X3Z, transition should be observable in the 33 000-cm-' region. The theoretical CASSCF/FOCI D of the ground state of Se2 was found to be 2.91 eV.lt2 A number of experimental De values were obtained from the predissociation of BO: bands (3.41, 3.16, and 3.10 eV).37
-
-
106 Chemical Reviews, 1990,Vol. 90, No. 1
Balasubramanian
TABLE 18. Spectroscopic Properties of Brza Photoionization and thermochemical studies seem to favor the higher value.171 Yee and Barrow170obtained Re ( d l i, ( c m - 1 ) Ass7gnment State an upper bound for De as 27096 + 2Xff,where 2X" is the Theory Expt. Theory Expt. Theory Expt. O+-lg splitting. They used a somewhat smaller value of 367 cm-I for this splitting compared to the theoretical 0 D 2.33 2.28 splitting and the observed splitting (Table 16). Balasubramanian corrected this by taking the 0;-1, splitting 2.81 2.66 : 3 3 G O '13220: as 512 cm-'. Balasubramanian thus obtained an upper 2.83 2.70 1C620 1 3 9 3 5 value for De as 27 608 cm-', still eliminating the highest of the three possible De values considered by these au2.61 2.68 11931 15903 thors (27 700,25 710, or 25 166 cm-I). Thus, the De value should be about 3.19 eV. The theoretical value of 2.91 3.21 3.li A8227 48930 eV seems to favor this more than the 3.41-eV value. A more accurate CASSCF/second-order CI followed by 3.19 49i7ti Davidson's correction for unlinked cluster correction 3.14 3 . 2 8 43530 53900 may yield a De value in closer agreement with the experimental results. 3.14 09633 Table 17 shows the weights of the leading configurations in the FOCI wave functions of the electronic 3.28 3.:: 49955 49929 states of Sez. As seen from Table 17, the 315;9, 'A , and l2: states are dominated by the 2ai17r4,17r2conhgura2.23 3,:7 53?-: 53:12 tion. The 3Au state observed in the 3A,-238- system arises from the 2uil7~3,17r39configuration. Tke A3n, 3.30 5?3!6 56337 state was found to be a mixture of the 2 a ~ 2 u , 1 7 ~ ~ 1 7 ~ ~ 3.15 6C 6 7 4 and 2 ~ ~ 2 a , l 7 r ~configurations. l?r~ The B32; state was found to be a mixture of the 2u~17r3,17ri,2ag2aula~17r~, . .J. 3 L E2EiC and 2ag2a,lx~17r4configurations. This seems to suggest that the C state ogbserved in the C32; system should be Theoretical spectroscopic constants for X(0:) were obtained configuration, although dominated by the 2ug2uUl7r~17r~ with the CASSCF/SOCI (4s4p2d basis) method. The theoretical D,(Br2) including spin-orbit effects = 1.87 eV compared to expersince this state is observed in the 53000-cm-' region, imental D,(Br2) = 1.97 eV. All theoretical constants are from ref contributions from Rydberg configurations could be181. Experimental constants and assignments are from ref 38. come significant. The spin-orbit contamination of the Of ground state '2:p'. The of Sez was found to be 81% 3Z; and i% TABLE 19. Spectroscopic Properties of Brzta O+(II)state is 74% '2' and 8% 32;. The 1, component State re ( A ) T, (cm-') w e (cm-1) 'II, was found to be actually 73% 311u Theory Theory Expt. Theory E x p t . ( la~la~2F~20,17r~17rg), 2 % 'nu (1u~la2,2a~20,17r~17rg), 18% 3ng ( l a 2 1 ~ 2 2 0 ~ 2 a , l n ~and l ~ ~ )0.2% , 'nu de
OF
(1u~la~2ug2u,1T,1Ag). $ 5
E. Br, and Br,'
i;
K2
There are numerous spectroscopic investigations on Brz. A summary of spectroscopic investigations up to 1977 can be found in the book by Huber and Herzberg.37 Brand and Hoy38have published a more recent review on the experimental spectra of halogens. The electronic spectra of Brz in the region below 60000 cm-'37B8reveal the existence of X, A, B, C, D, E, F, G, and H states. Among these, the A, B, C, F, and G states are observed in either absorption from the ground state or emission to the ground state. The D, E, and H states are observed in D B, E B, and H -,B transitions. The ground state X is assigned unambiguously to the lZ+ state. The A and B states are the 1, and 0: spinor%itcomponents of a 3nu state. The C state was assigned to the lII, state. The exact nature of these states (electronic configurations contributing to these states) and the other upper states was not clearly established from experiment alone, although the symmetry of the upper states is known in many cases from the intensities of these transitions. The B-X system has been studied by a number of workers. Rotational analysis, Franck-Condon factors, and the long-range potentials have been obtained for the B state by Barrow et al.178Ishiwata, Ohtoshi, and
- -
'q9
2N~
(3i:j
2.30
0
0
343
tl/zJ
2.30
3043
2820
342
A:
'-u
(3/2j
2.62
11 017
42
2;[L
(1/2j
2.60
14 117
3.04
20 312
128
3.35
20 584
127
3.58
20 718
89
2.79
25 033
199
8
'r; 'rIJ
(11)
%Ii (112)
c
1 6 414
376
194
190
137
132
"Theoretical constants are from ref 181.
Tanaka179identified an ion-pair state which they designated as an f state by a two-photon excitation technique. A transition from the ground state X to B followed by B to f was induced by using a two-photon technique. Subsequently, the fluorescence from the f to B state was detected. As these authors179pointed out, there was some controversy on the nature of some of the ion-pair states of Br,.
Heavy p-Block Dimers and Trimers
Chemical Reviews, 1990, Vol. 90, No. 1 107
Mulliken180derived the expressions for the energy levels related to the 311 and 'II states which enabled interpretation and assignment of the A-X, B-X, and C-X systems. Theoretical investigations before Balasubramanian's calculationslal were limited to the ground state of Br2.182J83Schwerdtfeger et a1.1a2employed a SCF/CI method with semiempirical pseudopotentials for the ground state of Br2. The calculation by Andzelm et al.183employed a single-configuration SCF treatment which tended to give a poor De. Balasubramanian carried out the first theoretical calculations on the excited states of Br2. Boerrighter et al.lMmade HF/limited CI calculations on the electronic states of Br2+. Balasubramanianls1 carried out CASSCF/FOCI calculations on 16 electronic states of Br2 and 6 low-lying electronic states of Br2+. Spin-orbit effects were taken into account by using the relativistic configuration interaction method which employed a double-l STO + polarization basis set for the bromine atom. The CASSCF/FOCI calculations were made by using a valence Gaussian basis set of (3s3pld) and (4s4p2d) quality. For the ground state, Balasubramanian also carried out full second-order CI calculations which included up to 210 880 configurations and tested the effect of spin-orbit interaction on the dissociation energies. The theoretical spectroscopic properties (Re,T,, we) of the bound states of Br2 calculated by Balasubramanian are shown in Table 18, and those of Br2+are shown in Table 19. The potential energy curves of some of the electronic states dissociating to the ground-state atoms in the absence of spin-orbit interaction are shown in Figure 6. The corresponding curves for Br2+ obtained with the FOCI method of calculation are shown in Figure 7. As seen from Table 18, the XO; ground state of Br has a theoretical FOCI Re value (3s3pld basis) 0.1 longer than the experimental value of 2.28 A. The larger basis set at the SOCI level of calculation yielded better results. The calculated FOCI we was found to be 10% lower than the experimental value of 325 cm-l, although the SOCI method improves this. Spectroscopic bands of Br2 observed below 60 000 cm-I were assigned to the A X, B X, and C X transitions. The A and B states are now well characterized as 311,(1,) and 311,(0:) states, respectively. The A and B states observed in the A X and B X transitions can thus be assigned unambiguously since the 0: state should be well above 311u(lu). The experimental Te values for the B and A states (15900,13 900 cm-') are consistent with the trend suggested by theoretical calculations. The experimental we values (153, 167 cm-') were found to be in very good agreement with the theoretical we value for the 311u state (155 cm-l). X absorption The C state was observed in the C system. It is also considered to be responsible for the predissociation of the B state. Although at the FOCI level of approximation the lIIU state appeared to be repulsive, higher order correlation effects and a larger basis set could lead to a bound 'II, state. Nevertheless, the C state must be relatively weakly bound since the absorption spectra that corresponded to the C X system are c o n t i n u o ~ s . ~ ~ , ~ ~ The D state observed in the D B system is most probably due to the 311g 311utransition. Since the B state is a 0: state, it was suggested by Balasubrama-
0 170
I
t >r
F
5
4 050
0 070
20
30
50
A0
R
- 6)
BO
,O
SO
Figure 6. Potential energy curves for the low-lying states of Br, (reprinted from ref 181;copyright 1988 Elsevier Science Publishers B.V.). See Table 18 for spectroscopic labels of known states.
x
- -
-
-
- -
-
1
2.0
3.0
5.0
4.0
6.0
R - 4 )
Figure 7. Potential energy curves for the low-lying states of Brzt (reprinted from ref 181;Copyright 1988 Elsevier Science Publishers B.V.). See Table 19 for spectroscopic labels of known states.
nianIal that the 0; and 1, components of the 311,(11) state are the most probable candidates for this transition. Since the 311 (11)state arises from more than a half-filled a shell, tke 2, and 1, states are lower than the 0; and 0; states, and consequently, the D state was assigned to 1,. The E state observed in the E B transition was assigned to the 311,(11)(0,+)state. The F state observed in the F X system could be due to the 3ZJ0,+) l2;(0:) or 311u(II)(O:) lZ,+(Og+) transition. The theoretical spin-spin effects reported in Table 18 for the 32; as well as the '2: state were obtained with a calculation that included only the 32; and 'Z: reference configurations and are thus crude. Ishiwata et al.179have studied the f B transition B f two-photon transition. These using the X authors call the f state an ion-pair state since the calculated Re value of the f state (3.17 A) is much longer than the Re value of the ground state. The f state was tentatively assigned to a 3Z;(Oa) state. It can be seen from Table 18 that the theoretical Re and we values of
-
--
-
-
-
-
108 Chemical Reviews, 1990,Vol. 90,No. 1
the 3Z;(II) state are in excellent agreement with the experimental values obtained by Ishiwata et al.179The observed T, value of this state is lower than the value reported in Table 18 since the 3Z;(II) state is split by spin-orbit interaction. The 0: component is lower than the unsplit 3Z, state. Thus, the f state participating in the f B system was found to be most consistent with the 0; component of the 3ZJII) state. The first 32; state was found to be repulsive. The G X system observed in the 56 337-cm-' region should be due to the G'Zi(0:) X'Z+(OP+)transition. The calculated Tevalues of 57 576 cm-5 in the absence of spin-orbit interaction was found to be most consistent with this assignment. The rotational analysis of this system should be carried out to confirm this assignment. The spectroscopic constants of Br2+ are shown in Table 19. Ionization of the highest occupied orbital of the ground state of Br2, namely, the lagorbital, yields the lowest state of Br2+as the Qgstate arising from the l a 2 1 a ~ 2 a ~configuration. l ~ ~ 1 ~ ~ There is another posskle low-lying state arising from the 1 u 2 1 a ~ 2 u 6 1 r configuration ~lr~ ( 2 2 i ) .Balasubramanianfs1carried out CASSCF/FOCI calculations that employed the (3s3pld) basis set on the 211g,211u,2Ag, %;, and 211u(II)electronic states of Br,+ which dissociated into Br+(3P)+ Br(2P). The ground state of Br2+is the 'II,(3/2) electronic state arising primarily from the l a ~ 1 u ~ 2 a 2 1 acon~lr~ figuration (see Table 19). The Re value o! the ground state of Br2+is about 0.08 A shorter than the Re of the ground state of Br2. The vibrational frequency is correspondingly larger (about 40 cm-l). There are no experimental Re values known for the electronic states of Br2+. Since the (3s3pld) basis set tended to yield Re values 0.1 A larger than the true values, Balasubramanianlsl predicted that the true R, values of the Q,(3/2,1/2) states should be 2.22 A. The experimental photoelectron spectra of Br, yielded substantially different w e values for the two components of the 211uexcited state.ls5 The reliability of these analyses was questioned in the literature. The we value deduced from the photoelectron spectrals5for 211,(3/2) is 190 cm-', while for the 211,(1/2) state it is 152 cm-l. Usually, the vibrational frequencies of the spin-orbit components of a A s state cannot differ that much unless there is considerable contamination of one of the components with another A-s state due to spinorbit coupling. As seen from Table 19, the theoretical vibrational frequencies of the two components of the first excited 211ustate are 194 and 137 cm-', supporting the experimental values deduced from the photoelectron spectra.185 The substantial difference in the two vibrational frequencies was puzzling. This difference arises from the interaction of the 211,(1/2) state with the 4Z;(l/2) and 22:(1/2) A-s states in the vicinity of its equilibrium geometry. These two states undergo avoided crossing, influencing the Re and we values of the TI,(1/2) state. Note that the Re is about 0.02 A shorter than that of the 211,(3/2) state. The 211u(3/2)component, however, is less contaminated, and thus its vibrational frequency is not substantially different from that of the 211ustate obtained without the spin-orbit term. The vertical ionization potential of Br2 at the FOCI level of theory was calculated to be about 9.55 eV. while
-
-
-
Balasubramanian
the adiabatic ionization energy was calculated as 9.47 eV. The ionization potential of the bromine atom calculated from the long-distance splitting of Br2(1Z,+)-Br2+(211g) was 10.45 eV compared to an experimental value of 11.8eV. Thus, the FOCI ionization potentials are 10% smaller compared to the experimental values mainly due to basis set and higher order correlation errors. When these values were corrected for the vertical and adiabatic ionization potentials of Br2, Balasubramanian obtained 10.51 and 10.42 eV, respectively. The vertical ionization potential calculated this way was identical with the photoelectron spectroscopic and photoionization value of 10.52 eV obtained by several a u t h o r ~ . l ~ ~ - ' ~ ~ The dissociation energy (De)of Br2+obtained for the 211gstate was calculated as 2.7 eV using FOCI calculations. Such calculations are typically 1615% in error compared to true values due to higher order correlation corrections and basis set limitations. If this value is corrected for these errors, a value of 3.04 eV is obtained, which was found to be in reasonable agreement with a D,,O value of Br2+estimated through the following relationship: Doo(Br2+)= D,O(Br,) + IP(Br) - IP(Br2) The value obtained from this formula and experimental data is 3.26 eV. The photoelectron spectra of Br2 were recorded by Cornford et al.ls5 as well as Potts and Price.ls6 The photoelectron spectra exhibited two states in the 19000-21600-~m-~ region and a third state in the 30 000-cm-' region. Since photoelectron spectra typically involve ionization of a neutral molecule, these bands must arise from vertical rather than adiabatic transitions. The theoretical vertical transition energies and the splittings of the photoelectron spectral peaks are compared in Table 20. The agreement between these two values is remarkable, which led Balasubramanian to reinterpret the photoelectron spectral bands as due to vertical rather than adiabatic transitions. As seen from Table 19, the Te values of these states are much lower than their Tvertsince the Re values of the excited states are much longer. Consequently, the earlier listing of these separations as (T,) (for example, see ref 37) must be corrected to Tvert. There is a discrepancy in the 211g(3/2)-211g(12) splitting obtained from photoelectron spectroscopy1E 4 and the convergence of the Rydberg series. VenkateswarlulMfit these series designated as R1 and R2 into the following formulas: Rl(vl) = 85165 - R / ( n - s ) ~
Rz(u2)= 88306 - R / ( n - s)'
n = 5, 6, ..., where the R1 series converged into the Br2+211g(3/2) state while the R2 series converged into the 211 (1/2) state. The 2 ~ , ( 3 / 2 ) - 2 ~1/2) g ( splitting calculate$ from the above formulas (3141 cm-l) is a bit high compared to the splitting obtained from the photoelectron spectra and Balasubramanian's theoretical calculations.lsl Balasubramanian's calculated splitting of 3043 cm-' is between this value and the value of 2820 cm-' obtained from the photoelectron spectra.ls5 Nonetheless, the theoretical spin-orbit splitting was found to be within
Chemical Reviews, 1990, Vol. 90, No. 1 109
Heavy p-Block Dimers and Trimers
TABLE 22. Contributions of Various Configurations to the Low-Lying Electronic States of Br, at Their Near-Eauilibrium Geometriesa
TABLE 20. Vertical Transition Energies for Brltn Transition
TVert (cm-') e x p e r i m e n t a1
theory 2ng(3/2) 211g(3/2) 2ng(3/2) 2r[
g
(312)
2ng(3/2)
+
+
+
+
+
2n , ( l / 2 )
3030
2820
'Fu(3/2)
18 170
1 9 290
2;[u(1/2)
21 478
21 602
zXi(1/2)
34 600
'i7"(11)
56 752
State
Conf ig u r a t i on (s )
2 4 4 2 4 4 20g In u I n g ( 9 4 % ) , 2 o u l n u l n9 (3%)
(30 700)b
Theoretical values are from ref 181. *The value is not certain.
TABLE 21. Spectroscopic Properties of the Ground State of Br2 at Various Levels of Theorya 0,
(evl
Method
Basis S e t
Re ( A )
FOCI
(3s3pld)
2.37
303
1.73
1.79
FOCI
(45'4p2d)
2.34
319
1.82
1.88
SOCI
(4s4p2d)
2.33
32 1
1.77
1* 87
e
we
(cm-')
Uncorrectedb CorrectedC
"From ref 181. *Without spin-orbit and Davidson's corrections. With spin-orbit and Davidson's corrections. 1-
the calculational error bars of both values. Table 20 shows the spectroscopic constants of the X ground state of Br, obtained by Balasubramanian using various theoretical methods. The simplest calculation employed the FOCI method with a (3s3pld) basis set, while the most sophisticated calculation used the SOCI method with a (4s4p2d) basis set (210880 CSF's). The uncorrected De is the value obtained from the appropriate calculations without the Davidson correction for unliked clusters and spin-orbit correction. The Davidson correction was estimated from the calculation by Schwerdtfeger et a1.,la2which yielded a De of 1.33 eV without the Davidson correction using a CISD method and 1.73 eV after the correction. The spin-orbit interaction decreased the De by 0.3 eV. Both these corrections were applied by Balasubramanian (Table 21). As seen from Table 21, the (4s4p2d) basis set yielded better De, Re, and we values compared to the (3s3pld) basis set. The improvement in De obtained by the SOCI method is marginal, although there are improvements in the Re and we values. The best De (1.88 eV) obtained by Balasubramanianlal compared quite favorably with an experimental value of 1.97 eV. The same calculation gave an we value of 321 cm-l compared to the experimental values of w e = 325 cm-' and Re = 2.28 A.37 The theoretical spin-orbit splitting of the bromine atomlal (2P3/2-2P1/2 splitting) from asymptotic splittings of molecular states of Br, was found to be 3915 cm-' compared to an experimental valuelN of 3685 cm-'. The curves in Figure 6, which do not include spin-orbit splittings, are actually split apart at long distances. The spin-orbit contamination of various X-s states became significant in this region in the RCI wave functions. For the ground state ' 2 ; the curve was found to be almost identical with the one reported in Figure 6 a t nearequilibrium distances in the absence of spin-orbit in-
)I3
2 4 3 20 20 i n I n [58:), 9 u " g
( 11)
2 3 4 2 ~ ~ 2 0 ~ 1 (33:) 1 i ~ 1 ~ ~
'From ref 181. The l u i l u i shell is omitted. For the 'II, state the contribution is a t 2.50 A.
teraction. At long distances the curve shifted downward to dissociate the 'P3/2 + ,P3/2 atoms. The ground-state atoms were found to be more stabilized by the spinorbit interaction than the ground-state molecule. The potential curves in Figure 7 for Br2+also did not include spin-orbit effects and were actually split apart by spin-orbit interaction. At the equilibrium the RCI spin-orbit splitting was found to be 3030 cm-'. The spin-orbit splitting of the Br+ ion 3Pstate into 3P2,3P1, and 3P0components was obtained by Balasubramanian et al.la9 using the RCI method in an investigation on the collision of Br+ with Kr. The 3P,-3P1 and 3P2-3P0 splittings of Br+ were calculated as 3182 and 4675 cm-l. Table 22 depicts the contributions of the leading configurations in the FOCI wave function of the various electronic states in the absence of spin-orbit interaction. As seen from Table 22, with the exceptions of 311g(II), 311u(II),3X&II), lz;, lnP(II),and lII,(II), other states have leading contributions with weights 185%. The 0' round state of Big was found to be 93% '2~(2ai17r,lag) 4 and 0.2% 311g(2a,17r&17r4).The 311,(0:) state was also found to be pure 311,(0$ at near-equilibrium geometries. The 311,(1,) state is 49% 31'Iu(2ai2a 17r417r3) 12% 111,(2a~2a,17r~17r~), and 7% 32:(2ag2u,17r,17rg) B 9 5 ' 3 a t 6.50 bohr. The 31'1,(2 ) state is 62% 311,(2a~2a,17r~17r~), 30% 311,(2ag2at17r,17r4), B and 9% 3A,(2a~2a217ri17r3).The O+(II) state is predominantly 3z;(o:U) (91.80); 1~:(0:u) states arising from 2ag2a,la~17r~ and 2o~2atl7ri17r~ make nonnegligible
4
110 Chemical Reviews, 1990, Vol. 90, No. 1
Balasubramanian
TABLE 23. Contributions of Various X-s Configurations to the Low-Lying States of Brr+a State
f-
P
z . ~ .
r: or,t r 1 b u t i on
State
Contribdtion
R (a.u.)
p:?! 6.00
1i 2 ,
3 'ZJ
4.75
1 ZU I
3 ,'2,
5.25
Z ~ * l n ~ l n ~ : ~ T ~ ( l 2. ~l )~ ,2 s ~ l n ~ 1 ~ ~ : ~ : ~ ( 5 8 ) ; g
u
S
5.50
OFrom ref 181. Note that the 211,(1/2) state exhibits an avoided crossing. The la$Tt contributions in percentage.
contributions (3%). The 311 (1,)state was found to be 311e (34%) and lII (30%), (20%) and lIIg (15%) arising from 2ag2a,l7r4,17r~at s 6.00 bohr. Consequently, spin-orbit contamination IS more significant in excited states than the ground state. The leading configuration contributing to the FOCI of the 211 ground state of Br2+ was found to be (loila32a!ln4,17ri (0.950) at 2.25 A. The 211ustate was predominantly (laila32a217r~17r~ (0.925), while the 211 (11) state was founi to be predominantly (l&o3ln:2o l7ri2a (0.902). The '2; and states arise from t8e (1u,1a32a22au17r~1n, Y configuration (85%) at 3.0 A. The 3Z.g+(fI)state was found to be predominantly 1ai1ae2agl7r4,ls;. Table 23 displays the contributions of various X-s configurations to the low-lying states of Br2+. As seen
hg
shell is omitted. Numbers in parentheses nre
from Table 23, the 211g(3 2) state is predominantly composed of the 2ai17rt17rg.configuration. At long distances, the 411g state arising from the 2ag2aulai17r~ configurations makes a substantial contribution. The 311 (1/2) component also exhibits a parallel behavior. $he compositions of the two spin-orbit components of 211u are interesting as a function of internuclear distance. The 3/2 component of 211uwas found to be predominantly 2 ~ ~ 1 7 r ~ l 7 r ~up ( ~ to I I ,4.75 ) bob. At 5.00 bohr, the 42;(3/2f state arising from the 2aK2a,17r~17r~ configuration makes a substantial contribution. Although this contribution is significant ( 11-16%), it is not large enough to induce an avoided crossing. The 1/2, component, to the contrary, exhibited a different composition. Up to R = 4.50 bohr, this state was found to be predominantly the 211u state arising frodl the
d
N
Chemical Reviews, 1990, Vol. 90, No. 1 111
Heavy p-Block Dimers and Trimers
TABLE 24. Dissociation Relationships for the Low-Lying States of In2 dissociation :imits
A-s s t a t e s
energies atomic s t a t e s
2P + *P
expt.a
("1) tbeoryb
0.0
0.0
dissociation l i m i t s u-u s t a t e s
a t m i c states
2 a ~ l x ~configuration. lx~ However, at 4.75 bohr, it became predominantly 4Z;(l/2) (50%) while Q,(1/2) makes a 26% contribution. At 5.00 bohr, the contribution of 211u(1/2)drops to 3% while 4Z;(l/2) increases to 61% accompanied by an increased 22:(1/2) contribution (7%), whereas at 5.50 bohr, the 211u(1/2) dominates once again. This avoided crossing of 4Z;(l/2) and 211,(1/2) explains the large difference in the vibrational frequencies (0,s) in the photoelectron spectra of these two states. After the author's manuscriptls1on Br2and Brz+went to press, he became aware of two investigations on Br2+.1&dJwThe laser spectroscopic investigation of the A-X system by Hamilto@' yielded a T,value of 16414 cm-' for the A state. If the A state is assigned to 211u(1/2), the agreement between the calculated T,and this value is good. Boerrighter et al.lMcarried out HF/limited CI calculation that included spin-orbit matrix elements in an empirical manner for Br2+. The author's R C P calcplations, which included single and double excitations from many X-s configurations, were in overall agreement with Boerrighter'slMfindings that the vibrational frequencies of the two components of the 211ustate differ due to avoided crossings. However, the author differed from Boerrighter et in the states that induce avoided crossings in that the author's calculations revealed it to be the 42;(l/2) state which undergoes avoided crossing with 211,(1/2). The QU(3/2) state mixes heavily with 42;(3/2) instead of 2Au-
energies ex2t.
(cm-1) tieor;
(3/2) as suggested by Boerrighter et al.lMbut does not undergo avoided crossing. The bond lengths calculated by Boerrighter et al.lMfor the 211ucomponents are also much too long.
F. In, The emission spectra of Inz were studied in a King furnace by Ginter et a1.lZ2and also earlier by Wajnkrac.lgl The investigation of Ginter et a1.122on In2 revealed emission bands in the 16 800-20 000-cm-l region centered at 18000 cm-'. Douglas et al.123studied the electronic absorption spectra of A12, Ga2, and Inz in cryogenic matrices. These authors found two systems, one at 13000 cm-' and the other centered near 27 690 cm-', for InB. Douglas et al.lB assigned the ground state of A12, Ga2, and In2 to a l2+ state and the absorption lZ; a n d k Z : X'2; transitions, bands to A'II, respectively. The theoretical investigation of Balasubramar~ian'~~ on Ga2 revealed these assignments to be incorrect and showed that the most consistent assignments of the bands observed by Douglas et for Ga2 are A311, X311uand B311 X311u(see section 1II.A). The dissociation energy ofIn2 was calculated as De 1 eV using mass spectrometric technique^.'^^ Froben et al.lg2recorded the Raman spectra of matrix-isolated Gaz, In2, and Ti2. They deduced approximate spectroscopic constants for the ground state of Inz and T12using the Guggenheimer method assuming
-
-
-
-
-
112 Chemical Reviews, 1990, Vol. 90,No. 1
Balasubramanian
TABLE 25. Spectroscopic Properties of In2 Calculated with the FOCI/RCI Methoda State
fe
(A)
T,
(Cn-')
ri
:C7l->)
0,
fE\')
3. :3
2.89
2-88 2.87 3.20
3.20
'
-004-
3.29
1
I
1
50
30
R
2.98
7 0
4 - 8
Figure 8. Potential energy curves of In2 obtained without the spin-orbit term using the CASSCF/FOCI method (reprinted from ref 194; copyright 1988 American Institute of Physics). See Table 25 for spectroscopic labels of known states.
2.97
3.29 3.2L
0.030r
?.Or 3.49
3.49
3.96
-
2.::
c
u) Q)
0.010-
c L
0
I
_-
v
:i:
g.-'
3.55
23 6 8 9
- "
3-
:. 33
aFrom ref 194.
1
W
0-
c I
- 0.0I O 1 a ground state of 32-for Inz. This estimate was subsequently found to be incorrect by Balasubramanian and Li.lg4 Balasubramanian and LiIg4 carried out CASSCF/ FOCI/RCI calculations employing RECPs for the indium atom which retained the outer 4d1°5s25p1shells explicitly in the molecular calculations. A (3s3p3d) valence Gaussian basis set was employed. The 4d1° shells of the In atoms were allowed to relax at the CASSCF level but no excitations from these shells were allowed. The CASSCF active space consisted of four al, two b2, and two bl orbitals in the CZugroup. RCI calculations194were carried out to estimate the spin-orbit corrections for the spectroscopic properties, dissociation energies, and potential energy curves. RCI calculations were carried out by using the orbitals generated by a SCF calculation of the 311uground state of Inz employing a double-r STO basis set. Identical RCI calculations were also carried out omitting the spin-orbit term. The differences in spectroscopic properties and energies obtained with and without the spin-orbit term were then applied as corrections to the corresponding CASSCF/FOCI properties. Table 24 shows the dissociation relationships for the low-lying states of Inz with and without spin-orbit
-0.020L
'
3.0
I
I
5.0
I
I
7.0
I
R-8 Figure 9. Potential energy curves of In2 including spin-orbit effects (reprinted from ref 194; copyright 1988 American Institute of Physics). See Table 25 for spectroscopic labels of known states.
terms. Also shown in that table are the theoretical and experimental asymptotic ~ p l i t t i n g s 'of ~ ~molecular states. As seen from Table 24, the agreement between the theoretical and experimental In atom splittings is very good. The spectroscopic properties of 23 electronic states of In2 are shown in Table 25.Ig4 The spectroscopic properties listed in that table for states without w-w components were obtained without the spin-orbit integral. The ground state of Inz is a 311,(O;) state as seen from Table 25. Figures 8 and 9 show the potential energy curves of electronic states of Inz without and with spin-orbit terms, respectively. The curves in Figure 8 were obtained by using the CASSCF/FOCI calculations. The
Chemical Reviews, 1990, Vol. 90, No. 1 113
Heavy p8lock Dimers and Trimers
TABLE 26. Some Allowed Electric Dipole Transitions and the Transition Energies for In$' Transition
Transition
T (cm-1)
T (cm-1)
1951
4321 26400
3ng ( I I )*3n
0: 3,-
+OL!
3r-
I
9
3
9,
- 5, 3
819 322 18156
550
25234 30323 29538
17653 29820 29035
.-_ I . .
45-
3533
'From ref 194. A two-way arrow indicates both states are bound. A one-way arrow indicates that the originating state is bound. The transition energies are adiabatic if both states are bound and vertical if one state is bound.
curves in Figure 9 were obtained by using RCI calculations that included the spin-orbit coupling accurately, but correlations were included to a lesser degree compared to the CASSCF/FOCI method. As seen from Table 25, the calculated Re, we, and De for the ground state (311u(O;)) of In, are 3.14 A, 111 cm-l, and 0.83 eV, respectively. The In-In bond strength in the 0; state is reduced by 19% due to spin-orbit interaction. The Re value is correspondingly increasing while we is decreased. The theoretical spectroscopic properties of the ground state of In, are in conflict with the values reported by Froben et al.lg2 using an approximate Guggenheimer method which resulted in an Re value of 2.8 A, which is too short for In2. The we value calculated by this method (115 cm-l) is, however, in reasonable agreement with the theoretical results of Balasubramanian and Li.lg4 The dissociation energy of 0.87 eV deduced by Froben et al.lg2 agreed with the Balasubramanian-Li theoretical value of 0.83 eV. The mass spectrometric methods yield a slightly higher De of 1eV. A similar discrepancy in the Re value of T1, was noted before. Pitzer and co-work-
ers195J96 obtained a bond length of 3.54 A for the ground state of T12,disagreeing with the 3-A value reported by Froben et al.lg2using the same technique. The experimental bands of the A X, B X, and the emission systems were centered at 13000, 27 690, and 18000 cm-', r e ~ p e c t i v e l y . ' ~ ~Table J ~ ~ J26 ~ ~shows both vertical and adiabatic transition energies of allowed dipole transitions of Inz. As seen from Table 26, the most consistent transition for the 13000-cm-' abX(311u(O;)) transisorption band is the A(311,(O;)) tion. The theoretical vertical transition energy of 12 423 cm-l is in very good agreement with the experimental band center (13000 cm-'). Since the 311 curve is repulsive, it is seen only in absorption. $he B X system was assigned by Balasubramanian and Lilg4to X(0;)) transition. The theoretical the B(311,(OJ) transition energy of 26400 cm-' was found to be in excellent agreement with the experimental value of 27 690 cm-l. The emission spectra observed by Ginter et a1.lZ2 (18000-cm-' region) are most consistent with the 3Z;(O:) 3ZJOl) transition. The theoretical Te of 18 156 cm-' found by Balasubramanian and Li was in remarkable agreement with this value; thus, Balasubramanian and Li assigned this to the 3Z;(O:) 32-(Oi)system. h a n y transitions listed in Table 26 have not yet been observed. Among the transitions in Table 26, only the above-mentioned three transitions have been observed. Thus, there is considerable room for further study in the electronic spectra of In2. There are no vibrational or rotational analyses done on the observed spectra. Specifically, the vibrational/rotational analysis of the bands in (3Z;-3Z:,) should be done for a comparison of theoretical Re and we values of these states. Spin-orbit effects were found to be significant for In2 since the atomic splitting of the indium atom is 2213 cm-'. The 311u(O;)-311u(2u) splitting of 1400 cm-' was about 64% of the indium 2P1/2-2P3/2atomic splitting. The spin-orbit splitting stabilizes the 311u(O:) state with respect to the separated atoms while it destabilizes the 311u(O;) ground state of In2 since the 311u(O:) state dissociates into In(2P1/2)+ In(2P3/2)atoms while the ground state (0;) dissociates into In(2Pljz)+ In(2Plj2). The influence of spin-orbit coupling on the bond lengths and the vibrational frequencies is subtle. The spin-spin splitting of the 3Z; state is much smaller (503 cm-l) compared to that of the 311ustate. Spin-orbit coupling also changes the shapes of the potential energy curves of some of the excited states of In2 For example, the double minima in the 0; and Oi(II1) surfaces are attributed to avoided crossings induced by spin-orbit coupling. Table 27 shows the contributions of various important configurations to the FOCI wave functions of the electronic states of Inz. As seen from Table 27, the 311u ground state and the 3Zi state are dominated by a single electronic configuration. The ' Z i and lZl(I1) states are mixtures of several electronic configurations. For the 'Z' state, both the 26: and 1 ~ configurations : make su6stantial contributions. Thus, correlation effects are somewhat significant for the lZZ states. The shoulder in the 3Z:(II) curve (see Figure 8) is due to an avoided crossing as seen from Table 27. At long distances this state was found by Balasubramanian and Lilg4to be
- -
-
-
-
-
-
114 Chemical Reviews, 1990, Vol. 90, No. 1
Balasubramanian
TABLE 27. Contributions of Important Configurations to the FOCI Wave Functions of I n i State
Percentage o f Contributions
State
P e r c e n t a g e of C o n t r i b u t i o n s
,
11
20 G
(91,
U
, 10u20ghu23u ( 3 \ , 1og20g20,:ng
( I I!
(l),
2 o u l n u (SO), 20 i n , ( i s ) , 9 3 l o u i n u (19) 3 a u l n , (4),4 o , l n ,
(4)
3 Zu(iI)
"From ref 194.
predominantly composed of the 2a 2a, configuration. At short distances, the contribution from the la,2ag17r~ configurations became important, resulting in an avoided crossing of these two states. The 32; state was found to be a mixture of 1 . ~ , 1 la,2ag17r~, .~~, and other configurations. Table 28 shows the contributions of various A-s states in the RCI wave functions of In,. The O;, O:, and 2, states are relatively pure (31'1,) while the 1, state exhibited 2 % triplet-singlet mixing. The most significant effect of spin-orbit contamination was found to be on the 0; states. The 0; curves exhibited avoided crossings. A t short distances, the O+(I)state was found to be predominantly l ~ i ( ~ Z whie ;) at intermediate distances, contributions from l . ~ i ( ~ Z ;and ) 2 ~ : ( ~ 2 .be3 came significant. At longer distances, the two states undergo an avoided crossing leading to a second minimum in the 0: surface. This avoided crossing is transferred to the OZ(I1) and Ol(111) roots, as seen from
the weights of the various X-s states as a function of internuclear distance (Table 28). Table 29 shows the Mulliken populations of various electronic states of In2 obtained from FOCI natural orbitals. As seen from Table 29, the 5s2shell is not an inert shell since Mulliken populations of In are between 1.79 and 1.84. The participation of the 5s2 shell in bonding is certainly of interest since for Tl, the corresponding 6s2shell was found to be inert.lg59lMThe 5s2 shell is not that inert for In2 since some of the 5s electrons are promoted to the 5p shell. The In-In bond is thus expected to be stronger than the T1-T1 bond, which is consistent with their calculated dissociation energies. Some of the excited states, especially the 3Z;, 311g(II), 3Au(II),and 3ZJII) states, arise from excitation of the 5s electron into the 5p shell as seen from their Mulliken s populations. The s populations for these states are 1.38-1.70. The In-In overlap populations are larger for the 32;, 311u,and lAg states.
Chemical Reviews, 1990, Vol. 90,No. 1 115
Heavy p-Block Dimers and Trimers
TABLE 28. Weights of Various A-s Configurations in the w-w States of Inf States
R (Bohr)
Percentage o f C o n t r i b u t i o n s
R (Bohr)
Percentage o f C o n t r i b u t i o n s
6.0
6.0
6.0
States
20
I n (3Tlu) ( 3 5 )
4 u
5.5
6.0 6.5
6.0 7.5 5.5 6.5 6.0
7.5
6.0 20 I n ( 9 9
5.5
6.0
1
"1
1 -
1 n u 1 n g ( E,)
3 +
l n u i n ( 2,) 9
5.5
6.0
5.5 6.5 7.5
6.5
6.0
'From ref 194.
G. Sn, The electronic spectra of Sn have been obtained by Bondybey and co-workerse7JeJ3iJcnas well as Nixon and co-workers.'98>200 Bondybey and co-workers have studied Snz using laser-induced fluorescence of matrix-isolated Sn2. The spectroscopic constants of the XO: ground state of Snz have been obtained. The strongest absorption for Sn, is the O:-XO+ system at 18223 cm-'. An emission system in the 12 600-14 000-cm-' region was assigned as terminating in a state 3000-5000 cm-l above the ground state. The spectroscopic properties of some of these states have been calculated from the electronic spectra. Balasubramanian and Pitzerml carried out SCF/RCI
calculations including spin-orbit coupling on ten lowlying electronic states of Sn2employing a double4 STO basis set. The partition function of Sn, was also calculated with the objective of correcting the thermodynamic Doo value of Sn,. Unfortunately, in the final correction term, these authors had wrong sign, which was subsequently corrected in an erratum. After the paper by Balasubramanian and Pitzer201 appeared, Pacchioni202calculated the low-lying electronic states of Sn, and Pb,. He used semiempirical pseudopotentials and the MRDCI method. Table 30 shows the dissociation limits for low-lying molecular states of Sn2 and the energies of separated atoms. The tin dimer has many more low-lying electronic states compared to Ge,, mainly due to larger spin-orbit coupling.
116 Chemical Reviews, 1990, Vol. 90,No. 1
Balasubramanian
TABLE 29. Mulliken Population Analysis of Inz" ...._
State
In
Ir(s)
In(p)
-
.
uver aD
bRUSS
Il(dj
il-I1
TABLE 31. Spectroscopic Constants of Several Low-Lying States of Sn, and the CorresDondina - Values Obtained without the kpin-Orbit Term"
13.000 X
::.300
0
2.76
157
342
2.75
205
: 577
2.62
218
2 509
2.62
220
7 159
2.81
178
4 084
2.63
219
4 192
2.63
221
250
2.52
232
O+
9
13.000
1
13.000
2,
13.300
'U
:3.900
29
13.000
0;
:3.ocIc :3.900
0: 1"(II)
:3.000
lg(!I)
9 354
2.95
a8
:3.oc0
Oj(I!)
8 002
2.78
li6
OpI)
i a 223
.- 2_. 3 3 c
31;
1 594
2.75
294
.. -2.lrJC
3, U
3 213
2.52
2:9
:3.300
:-*
7 93:
2.81
140
-5
t
:?.ZOO
-a
7
"From ref 201. The T , value of the O:(II) is an experimental value from ref 140.
::.330 :3.330
.. 2_, 9 3 c ::.SO0
"The population analyses were carried out at Re if the state is bound; otherwise they were carried out at the Re of the ground state. From ref 194.
TABLE 30. Dissociation Limits of Some Molecular States of Sn, Dissociated
Energies of dissociated d t m s
Mo: ecu' a r
stat es
atoms
in 0 - l Snza
Table 31 shows the theoretical spectroscopic constants for Sn2 calculated by Balasubramanian and Pitzer.201Figure 10 shows the potential energy curves of the g states of Snz, while Figure 11 shows the corresponding curves for the u states of Sn,. The vibrational frequency of the ground state of Snz is well-known as 188 cm-' in rare-gas matriceslBP2@' and 190 cm-' in the gas phase.lg7 The theoretical value 197 cm-' agrees well within the estimated uncertainty of 10 cm-'. Most of the excited electronic states that have been observed in Sn2 lie above the 10000-cm-' level of the highest state among those calculated theoretically. The first 0: state of Snz is at 4192 cm-l, which appears not to have been studied experimentally. The strongest absorption in Snz is to a second 0: state a t 18223 cm-' from the ground state. An emission in the 12 000-14 000-cm-' range for Sn, was interpretedlg8 as leading to a state about 3000-5000 cm-l above the ground state with a harmonic vibration frequency about 195 cm-'. Both the 0: and 0; states near 4100 cm-' fit the first criterion, and the 1, state at 2500 cm-' is probably close enough to be considered. These are all components of the 311uterm in A-S coupling. Their theoretical 0,s are a bit too high, near 220 cm-'. Since these bands are relatively broad and both the vibrational assignment and the anharmonicity are uncertain, it is not clear if the experimental w,s are definitive. Balasubramanian and PitzePl suggested the possibility that this array of bands contains contributions from more than one transition involving the ,:O O,; or 1, states. The thermodynamic "third-law'' method204~205 of calculation of De of Snz from mass spectrometric data requires knowledge of the partition function of the molecule and the dissociated atoms. The molecular
Chemical Reviews, 1990, Vol. 90, No. 1 117
Heavy p-Block Dimers and Trimers
where Aci is the excitation energy above the ground state (A€ = Te),gi is the electronic degeneracy, qrotand q&,r are the rotational and vibrational partition functions, and the s u m is over all electronic states that make an appreciable contribution. The lowest electronic state usually makes the predominant contribution to the partition function denoted by qo, which can be factored out. Thus, the above expression for q can be rewritten for a diatomic molecule as
R
(a)
Figure 10. Potential energy curves for the g electronic states of Snz (reprinted from ref 201; copyright 1983 American Institute of Physics). See Table 31 for spectroscopic labels of known states.
2ot
1
3P
+ 3P,
3P0+
'Po
In the above expression, the vibrational factor is approximated by that for a harmonic oscillator. Balasubramanian and Pitzer201calculated q for Sn2 using o,,~ = 189 cm-' and the theoretical spectroscopic constants in Table 31 for other states. At 1600 K the sum in q above was found to be 2.93, which was only a bit smaller than the value 3 assumed before on the basis of a 32, ground state. A much greater change arises, however, from the reduction of we,o from the value of 300 cm-' assumed beforem*% to 189 cm-l. The change in Re is smaller but also significant. The mass spectral data* were presented in the form of Doo values for a many temperatures from 1472 to 1769 K calculated on the basis of a partition function with defined parameters. The correction to each Doo value can be shown to be 6 D t = RT In (q'/q'') where q was the originally assumed partition function and q I f is the corrected value. At 1600 K, the ratio q " / q is 1.43, which yields a 6Doo of -1.15 kcal mol-'. The corrected D: is 1.94 eV, in excellent agreement with a theoretical value of 1.86 eV. H. Sb,
Electronic spectra of the antimony dimer (Sb2)have been studied by many investigators. In summary, Genard206first studied the fluorescence spectrum of anR (d) timon vapor, which revealed four bands in the 2968Figure 11. Potential energy curves for the u electronic states 3132- region. NaudG207observed two bands, which of Sn2(reprinted from ref 201; copyright 1983 American Institute were assigned to the D X'Z: and F X'Z: systems. of Physics). See Table 31 for spectroscopic labels of known states. Nakamura and Shidei208showed the existence of anparameters of Sn2 were estimated in this calculation. other system. Almy and Schultz209found two other Spectroscopic m e a ~ u r e m e n t s ~ ~have J ~ ~provided J ~ ~ - ~ ~ systems, assigned to A XIZ+ and B X'Z'. Mrovibrational frequencies in good agreement with the zowski and Santaram210studiea the D X'Zf system Balasubramanian-Pitzer theoretical calculations. Bain emission. Sfeila et a1.211carried out the vibrational lasubramanian and Pitzer201corrected the De obtained and rotational analyses of the B X bands observed from mass spectrometry using the correct partition in emission, which provided the spectroscopic constants function. This method is described below. A similar for the B and X states. Topouzkhanian et al.212p213 correction to the De of Pb2 was made by P i t ~ e r . ~ ~ ~ carried out the vibrational and rotational analysis of the In general, the partition of a diatomic molecule can D X systems. be separated into translational, rotational, vibrational, Laser-induced fluorescence spectra in rare-gas maand electronic factors. While the translational factors trices have been recorded by Gerber and Kuscher214as are the same for all electronic states, the rotational well as Bondybey and c o - ~ o r k e r s . Sontag ~ ~ ~ and factor depends on Re (or Be) and the vibrational factor Weber216v217have also investigated the laser-induced on we. Hence these factors should be considered sepafluorescence and Raman spectra of Sb2 and Sb4 in rately for each electronic state. Consequently, the rare-gas matrices. All these investigations have resulted complete partition function for a diatomic molecule can in accurate determination of the spectroscopic constants be expressed as of the X, A, and B states. Bondybey, Schwarz, and Griffiths215 observed 4 = 4tr(T,P)Cqr,,(T,Re,i)q~~(T, we ,Jgie-AfilkT i fluorescence bands in the 15000-cm-' region, tentatively 2.5
3.0
I
I
3.5
4.0
-I
K
-
-
-
-
--
118 Chemical Reviews, 1990, Vol. 90,No. 1
assigned to a 3Z:-X'Z: based on the similarity of these bands to the Vegard-Kaplan system of N2. Balasubramanian and Li219have shown that this assignment is not consistent with the theoretical separation of the state from the ground state and reassigned these bands to the 3Au(l,)-X1Z,f system. Mass spectroscopic investigations of antimony clusters also have been carried out by Kordis and Gingerich.218The vacuum-UV spectra of Sb2have been recorded.147 Balasubramanian and Li219carried out relativistic complete active space MCSCF (CASSCF) calculations followed by large-scale configuration interaction and RCI calculations on 33 electronic states of Sbz lying below 44000 cm-'. An extended (4s4p3d) basis set that included polarization functions was employed. The outer 4d1°5s25p3shells (15 electrons) were explicitly retained in these calculations. The rest of the core electrons were replaced by the relativistic effective core potentials. CASSCF and CI calculations of all the electronic states were done in the C2ugroup, and a few states were also calculated in the DZhgroup to compute the effect of higher order correlations. The DShsymmetry was chosen to facilitate the inclusion of higher order excitations since in the Da group the configuration counts are much smaller compared to the same calculations in the CPugroup. Following CASSCF, CI calculations were carried out with the first-order CI (FOCI) approximation. The CASSCF/FOCI calculations, although they appeared to be satisfactory for calculating the Re and we values of various electronic states, were found to be somewhat less accurate for energy separations of electronic states of Sb2. These calculations yielded only 70% or less of the experimental dissociation energies, especially for group V dimers, for which correlation effects were found to be quite severe. Hence, the effects of higher order excitations not included in the CASSCF/FOCI calculations were studied by the CASSCF/MRSDCI methodology for a few low-lying electronic states of interest. MRSDCI calculations of Sb2 included single and double excitations from a set of chosen reference configurations that had a coefficient 10.07 in the CASSCF. The CASSCF calculations included up to 400 configurations, while the MRSDCI calculations included between 60 000 and 300000 configurations. Balasubramanian and Li219also carried out RCI calculations to estimate the spin-orbit effects. The RCI calculations were carried out by using the recently developed RCI method for polyatomics which employs natural orbitals obtained from a CASSCF/CI calculation and Gaussian basis sets. The RCI calculations included possible low-lying A s states that had the same R symmetry and potentially mix in the RCI. Table 32 shows the spectroscopic constants of 33 electronic states of Sb2 including spin-orbit effects obtained with the CASSCF/FOCI/RCI methodology. The calculated FOCI potential energy curves are shown in Figures 12 and 13 for the singlet and triplet states, respectively. Figure 14 shows the potential energy curves of Sbzwhich included spin-orbit effects. As seen from Table 32, the spin-orbit components of a given A-s state came out in a group. That is, the spin-orbit contamination for Sb2 is not so large as to cause a
Balasubramanian
TABLE 32. Spectroscopic Properties of Sbz Calculated with the FOCI/RCI Method' state
Re Theor.
(a) Expt.
T,
3
-1;
2.59
3
D,
(ev)
Theor.
Expt.
Theor.
246
2'3
:, 86
836
245
1.85
7523
177
0.93
7910
175
3.92
8439
175
0.9:
:6336
199
:.E3
16388
2.46
Expt.
0
2.34
ye ( c m - 1 )
(cm-1)
Theor.
:4991
I97
217
:.c2
15512
199
:.53
16505
197
i.58
:?88?
2:;
.~. .. . .
20439
:sa
2:052
2: 3
3.30
2:525
2;:
3.:3
??ala
2:L
0.95
:::-e
...J3: o-
L "
2c5
r.."
1
;:3
:,53
22625
?:5
:.55
2226C
205
:.58
21370
205
2.44
22976
206
2.C6
23472
140
0.72
23496
140
0.72
26158
193
2.06
26614
211
2.57
27570
218
1.89
31319
143
1.42
33525
:5l
1.15
33583
:*a
36154
3:430
1Cl
35243
32033
i59
%
I ,
-9
'2.36
239
!.T$
3-156
!5?
C.'?
39348
3 .
..6
P ,.-I
i2:30
85
3."
'
20
__I
-
I.
-" 3i
i.
"Theoretical constants are from ref 219. Most of the experimental values are from ref 137 (see text).
Chemical Reviews, 1990, Vol. 90, No. 1 119
Heavy p-Block Dimers and Trimers 0. 0.10. 0.08
-
0.06. u)
2
0.04-
,
1
2.0 3.0 4.0 5.0 6.0 7.0 6.0 R
4
Figure 12. Potential energy curve8 of the singlet electronic states of Sb2 (reprinted from ref 219; copyright 1989 Academic Press, Inc.). See Table 32 for spectroscopic labels of known states.
-0 0 5 1 -0 061
"q* 1P$O$
-0 07
0.14-
20
25
30
35
40
45
50
55
R 4 H
5' +'P u)
w
-0.02.
2.0 3.0
4.0 5.0 R
6.0 7.0 0.0
+i
Figure 13. Potential energy curves of the triplet electronic states of Sbz (reprinted from ref 219; copyright 1989 Academic Press, Inc.). See Table 32 for spectroscopic labels of known states.
change in the relative ordering of the electronic states. The ordering of the spin-orbit components of most of the low-lying states of Sb2 followed Hund's rule with the exception of the 3A,(l,) state. The contamination of 3Z:(l,) with 3Au(lu) lowered 3Z:(l,) and raised 3Au(1,) relative to the 3Au state. The spin-spin and spinorbit splittings of the and 31;; states were found to be smaller compared to the 3Au and 3Z; states. Similarly, spin-orbit mixing of 3A,(2,) with 'A,(2,) lowered the 3A,(2,) state. Table 33 shows the spectroscopic constants of nine low-lying electronic states of Sb2 obtained with more accurate CASSCF/MRSDCI/RCI calculations.219 In comparing the results of Tables 32 and 33, one can see that Re shrinks by 0.01 A due to higher order correlations. The vibrational frequency improved by 4.5% for the ground state. The main effect of higher order correlations not included in the FOCI was found to be
":
Figure 14. Potential energy curves of the low-lying electronic states of Sbz (reprinted from ref 219; copyright 1989 Academic Press, Inc.). See Table 32 for spectroscopic labels of known states.
on the dissociation energies. The calculated Des increased by 17% due to the higher order correlations. The refined theoretical De (2.17 eV)219was still about 30% smaller than an experimental (thermodynamic) value of 3.09 eV obtained by the Knudsen effusion mass spectrometric method.218 The Re and we values of the X ground state deduced from laser-induced fluorescence experiments are 2.48 f 0.02 A and 269 cm-l. The best levels of calculations (MRSDCI) of Balasubramanian and Li219yielded 2.58 A and 259 cm-'. The slightly longer bond length is attributable to the effective core potential approximations by Balasubramanian and Li.219 The longer bond lengths and slightly lower vibrational frequencies appear to be the general trend in the effective core potential calculations and are consistent with other calculations on comparable systems. Table 34 shows the dipole-allowed transitions for Sb2 and their adiabatic transition energies. As seen from Table 34, there are 31 allowed electronic transitions. Among the transitions listed in Table 34, only four to five transitions appear to have been observed, all from the ground state, and only two of them were experimentally well characterized. Two experimental systems labeled B-X and A-X, where X is the 'Z,'(O:) ground state, have been studied to a considerable extent. The Te value of the B state deduced from the experimental spectra is 19OOO-19070 cm-'. Balasubramanian and Li219 calculated the CASSCF/MRSDCI/RCI separation of 3Z;(O:) from the X(0:) ground state to be 20255 cm-l, in reasonable agreement with the experimental results. The oevalue of the B state deduced from experiments was found to be 218 cm-'. The small discrepancy between the theoretical value and the experimental result is about the
120 Chemical Reviews, 1990, Vol. 90, No. 1
Balasubramanian
TABLE 33. Spectroscopic Constants of Various Low-Lying Electronic States of Sbz Calculated with the MRSDCI/RCI Methodn
( lu)-lFl(Oi) transition. The theoretical adiabatic transition energy of 15 636 cm-l is in very good agreement with the observed value of 14 995 cm-'. Further, the theoretical Re and we values when corrected for the ECP a proximation and electron correlation (AI?,= -0.17 Po, = +18 cm-') yield 2.60 A and 216 cm-'. These values are in remarkable agreement with the experimental values of 2.64 A and 217 cm-'. Further, there is no other allowed electric dipole transition in this region. Bondybey and co-workers215noted that the fluorescence in the 15 000-cm-' region exhibited an intensity ratio of I , , / I L= 1.32. This ratio is close to a An = +1 intensity ratio of 4/3 = 1.33, suggesting the A state to be a 1, state. However, they assigned the A state to 3E:(l,) based on the Vegard-Kaplan system of N2 and Asz. But the separations of the electronic states of Nz are quite different from those of Sb2,and thus it seemed that it is not entirely correct to assign the observed transitions of Sb2based on N2 or even As,. The 3Z:(lu) state of Sb2is much lower (T, 8760 cm-') in energy and thus did not appear to be a probable candidate for the A state seen in the region of 15 000 cm-l (see Table 33). Moreover, the 3Z: state has a slightly longer bond length (2.79 A) and much lower w e value (177 cm-l). Even if these values are corrected for the ECP approximations, the theoretical Re and we values of the 3Z:(l,) state are 2.65 and 195 cm-l. The experimentally observed A state had a higher o,value of 217 cm-l. All these reasonings together with the energy separation of 3A,(l,) led Balasubramanian and Li219to speculate that the A state observed in the A-X system is most probably the 1, component of the 3Au state. The experimental T, and o,values are uncertain for the D-X system observed in the 32087-cm-' region. Theoretical calculations of Balasubramanian and Li219 suggested (see Table 34) three candidates for the D state (3rIu(II),E ' :, and 'II,). The vibrational frequencies of the 311u and lIT, states are somewhat smaller compared to an uncertain experimental we of the D state (209 ~ m - ' ) .The ~ ~ most probable candidate for the D state was believed to be the '2: state since its we value should increase by at least 15 cm-' due to higher order correlations (184 cm-l). The predicted theoretical we value of the D state after correction for the ECP approximation (202 cm-l) was found to be in remarkable agreement with an uncertain experimental value of 209 cm-'. Balasubramanian and Li assigned the observed D-X system tentatively to the D12:-X1Z; transition. Sontag and Weber216 identified a new electronic system, which they called K-X, in the 32 000-cm-' region below the D state. The vibrational and rotational constants of the K state have also been determined. The Re and we values deduced from laser-induced fluorescence experiments are 2.841 A and 127 cm-l, respectively. From Tables 33 and 34, the most consistent candidate for the K state is 311,(0:), arising from the second root of the 311ustate. The theoretical Re and we values (2.96 A and 141 cm-') are in reasonable agreement with the experimental values, although the experimental values themselves are not very definitive due to perturbations with nearby electronic states. carried out the rotational analysis of the Sibai et D X system. They found that the vibrational levels of the D state were quite perturbed by another electronic state in this region designated as the L state. As seen from Tables 32 and 33, there are many probable
1,
--
2.58
836
258
2.16
3+IL;
2.78
8760
:99
1. 3 4
AT+
3
-
- ::")
:"-
31-,
1Ad[2uj A
*
IJ
2.75
2":'"
?C5
2.c3
2.75
20339
205
2.02
2.73
23509
234
2.72
25i:j
2.86
33945
I*_-
36296
__ __
232
32087
180 16:
209
--
__
nFrom ref 219.
same for both the ground state and the excited states. The predicted difference in the o,values between the X and B states (54 cm-l) was found to be in excellent agreement with the experimental difference of the o, values of the X and B states. Likewise the theoretical difference in the Re values of the X and B states (0.17 A) was found to be in full agreement with the experimental difference of 0.17 A. This appeared to substantiate the assumption that the small discrepancy between the theoretical and experimental results is mainly due to the effective core potential approximation. The calculations of Balasubramanian and Li219 confirmed the earlier assignment of the B state to the 0: component of 32- Bondybey et al.215. Bondybey et a1.215u as well as Sontag and W e b e P studied the A-X system both in the gas phase and in a rare-gas matrix. The T , and we values of the A state deduced from these experiments are 14990 and 217 cm-', respectively. The Re value of the A state has not yet been obtained from the gas-phase experiments, but in the Ar matrix a value of 2.64 A has been suggested. As seen from Table 33, the most consistent electronic transition for the A-X system with the theoretical calculations of Li and Balasubramanian is the 3.1,-
-
Chemical Reviews, 1990, Vol. 90, No. 1 121
Heavy p-Block Dimers and Trimers
TABLE 34. Some Allowed Electric Dipole Transitions of Sbz and Their Transition Energies'
-
T (cm-') Transitions
1
serve6
T (cm-') Transitions
FOCI/RCI
7823
8760
16388
15636
A 14955
19747
22176
20255
B 19070
23496
22435
20513
36154
36296
K
36243
33945
D 32087
MRSOCI/RCI
Observed
18335
25702
L 31400
37156
25760 31225 34324
41024
F 44433 ?
11977
44492
H
44730 ?
13142
42150
15562
12064
15586
:2515
19550
:3229
234J9
:3632
3;138
15549
34737
15673
"ram ref 219.
candidates for the L state. Since the K and L states are close in energy and are both below the D('Z:) state, Balasubramanian and Li219tentatively assigned the L state to 3rIu(1u). A series of absorption bands labeled as the H X and F X systems were observed at 44 400 and 44 760 cm-l, respectively, by Topouzkhanian and co-workers.212 Balasubramanian and Li's calculations are consistent with these observations in that there are at least three allowed electric dipole transitions in the 44 000-cm-l region. The 311uand 3Z:(II) states are repulsive while the 111,(11)state is bound. It was not evident which one of the three possible candidates corresponded to the H and F states. Balasubramanian and Li219argued that the experimental (uncertain) we value of the H state (479 cm-') is unreasonable for a heavy species such as Sb2. As seen from Table 34, there are 22 electronic transitions predicted by theoretical calculations that appear to have not yet been observed. With the exception of the 32:(lu) X'Z; transition, all other transitions involve excited electronic states and thus would depend upon the lifetimes of these states. The
-
-
-
3Zi(lu)-X121(Oi) transition is allowed in the perpendicular direction (An = l ) , and this should certainly be observable in the 8000-cm-l region. If this state is detected, the assignment of the A state to 3Au(lu)by Balasubramanian and Li219will be confirmed. The 2, component of the 3Austate could also be observed with multiphoton methods. Table 35 depicts the weights of various electronic configurations in the FOCI and RCI wave functions of the electronic states of Sb2. As seen from Table 35, electron correlation effects are quite important even for the ground state of Sbz. The weights of the leading configurations are >80% for the 3Au,32;, 'A,, 329',3Ag, l2;, lII , 32;,lZ:(III), lAg, 311u(II)7 In,, and 311ustates. The l2: state is the most complex mixture in that three spatial configurations contribute to a great extent for this state. Table 35 also shows the spin-orbit contaminations in the RCI wave functions of the electronic states of Sb2. The mixing of the l2; state with the 311g(0,+) state was found to be only 2% for the ground state at its equilibrium geometry. This is also reflected in the relatively small lowering of the De of Sb2 by the spin-orbit term
122 Chemical Reviews, 1990, Vol. 90,
No. 1
Balasubramanian
TABLE 35. Contributions of Important Configurations to the FOCI Wave Functions and the A-s Contributions to the o-o States of Sb2' 2
4 1
2 0 In ( 2;) g u
2 2 2 1 ( 8 2 ) , 2 0 I n I n ( 2;) g
u
g
(lZ),
1 4 1 3 20 I n I n ( TT ) ( 2 ) g
u
g
9
20 2 I n 4 ( 7 9 1 , 2 o i l n ; l n i 9
23
(10)
u
2 g
In
3 u
In 1 ( 3A,,) g
(3)
2 g 2, g L n3u l n1g (3Z,J+. ( 8 5 ) , 2 2~ ~1 3l n3 Z u~1) l ( 1~2 ) ~ (
2~
2 g
In3 ' 1 (3 b u ) ( 8 4 ) , 2 a 2 In 1IT'( 7 3 A J ) (8), u g g u g
2 3 ' 1 1 2 0 I n I n ( "1 g u g
(5)
2 3 1 3 2 o g l n u l n g ( Tu ) !9 5 )
"From ref 219. Complete l u g and lo, shells are not shown.
since decrease in De is brought about by the bondingantibonding mixing as a result of contamination with the 311gstate. The stabilization of the 1, component
of the 32: state is primarily due to the mixing of 32:(1,) with 3Au(lu). Similarly, the 3A,(2,) is stabilized by mixing with 'A,(2,) (about 5%). Thus, the conventional
Heavy p-Block Dimers and Trimers
Hund's ordering of 3Au spin-orbit states (lu,2,, 3,) is violated since 2, is lowered by contamination with la,. Table 36 shows the gross and overlap Mulliken populations of 22 electronic states of Sb, obtained from the FOCI natural orbitals. As seen from Table 36, the gross s populations of most of the electronic states are 3.0. The small increases in the d populations were primarily a consequence of the contribution from the d polarization functions. Also reported in Table 36 is the population analysis for l2: a t a long distance (6.5 A) to comprehend the effect of atomic correlation in the population. The effect of relativistic mass-velocity contraction on the stabilization of the 5s shell is of interest. For the next row, it is known that the 69, shell is so stabilized by the relativistic mass-velocity effects that it does not participate in chemical bonding (the "inert-pair'' effect). The gross s populations of most of the states are smaller than 2.0, indicating the participation of the 5s shell in the bond. The ground state has the largest overlap population (0.88) since the Sb-Sb bond is composed of a triple bond. It is interesting to note that the overlap population of the comparable In, molecule194in its ground state is 0.335, implying that the Sb, bond is much stronger than the In,. The ratio of the two overlaps could approximately be taken as the ratio of the bond orders (2.62). The negative overlap population in the '2: state appears to be primarily due to mixing of the 2ag2auconfiguration in this state, in which the 2a, orbital is antibonding.
Chemical Reviews, 1990,Vol. 90,
TABLE 36. Mulliken Population Analysis of Sbz"
-
-
overlap
ross
state
I . Te, The electronic spectra of Te, have been the topics of many s t ~ d i e s ~ for ~ ~several i ~ ~ years. * ~ ~ The ~ electronic spectra of Te, are of considerable interest since they serve as possible wavelength standards. Further, the Te2 molecule has been considered as a candidate for optically pumped lasers. Theoretically, the Te2 molecule is considered interesting as a result of relativistic effects and the complexity of electronic states due to large spin-orbit coupling. The strongest observed transitions of Te2 are the B X and A X systems. These transitions were observed in both absorption and emission spectra as well as laser-induced fluorescence and chemiluminescence spectra. Yee and Barrow170observed a weak fluorescence serive originating from the perturbed B(0:) terminating to a state X,
i n
3.33
2 : 636
2 2 20:e 2 2 1553
131
*-
22 222"
cc
4::
Li,
At2
A
:.E3
j?:?
222
2.6e
6900
235
_ -. - . ,I, , ,
9::3
;E:
2 3-.
:3 3G:
1;-
~
.* 3:i .
2 . ?E
I
__ :35
. a 52'-, c
2 2 4:3
132
3.5;f
25 7 2 2
92
3 . ;if
2: 2 3 3
53
3.4Zf
2 i 547
91
3.05'
34 0 c 3
147
3 . 4Zf
37 135,
84
3.33
"Reference 37. *Reference 232. 'Predicted value as in ref 234. This state is yet to be observed. dReference 230. eReference 231. fcalculated Re and we values for these states may not be accurate due to basis set limitations. gAll theoretical constants are from ref 236.
and R a v i m ~ h a together n ~ ~ ~ with available experimental further with more diffuse functions and polarization functions. values. The potential energy curves for several of these states are given in Figure 15 (without spin-orbit couAs seen from Table 37, theoretical calculations confirmed the earlier assignments of the A(O:)-X(Of) and pling) and Figure 16 (RCI, including spin-orbit couB(O:)-X(Ol) systems. The A(0:) state was founA to be pling). As seen from Table 37, the ground state of Te, is an 311,(0:) while the B(O:, state was found to be 3Z;(O:). X(0,f) state. The 1, state, which is the other 9 comThe A" state observed by Bondybey and EnglishZ3lin ponent of the 3 X ; X-s state, is 2229 em-' above the 0; the weak fluorescence, which was assigned to 311,(2,) state. The theoretical T , of the 1, state was found to by these authors,231is most consistent with 34u(1u) be in very good agreement with the value of 1975 em-' (Table 37). The theoretical Tevalue of 17 759 em-' was reported in ref. 13 and 14. The calculated Re, T,, and found to be in very good agreement with the experiwe values were found to be in very good agreement with mental state. The notation A' was also used for another available experimental data for the electronic states electronic state with a T, value of 14091 cm-l by other below 10000 cm-'. The theoretical Re values of excited authors, and thus caution must be exercised in talking states above 10 000 em-' are much longer than the exabout this state. The state that Bondybey and Engperimental values. This difference was attributed lishZ3ldesignate A is observed in the A' X?(1,) mainly to the basis set limitations in the theoretical fluorescence. The lower state is assigned to 1 since calculations by Balasubramanian and R a v i m ~ h a n . ~ ~excitation ~ to the A' state could not be achieveA from For the excited states the basis sets must be extended the X(0:) ground state. Balasubramanian and Ravi-
-
Heavy p-Block Dimers and Trimers
Chemical Reviews, 1990, Vol. 90, No. 1 125
TABLE 38. Leading Electron Configurations Contributing to Some Low-Lying A-s States of Te2 near Equilibrium Geometrya
"
O 0.06 .O8I
-
-
2 4 2 2 2 4 20 I n I n (go%), 20 I n n ( 3 % ) P u g g u g
-
0.04
1
Q Y)
2
-
0.02
39
-
c
7
2
4
2
2
2 4
2
4
2
2
2
20 I n I n ( 8 3 % ) , 2 0 In n (5%) g u g 9 u g 4
2 0 In I n [ 7 8 % ) , 2 0 I n I n ( 1 4 % ) g u g 3 u g
0 -
W
-
3Au
-
31;
2og I n u Ing[ 90%)
31-
20
-0.02
-0.04
'
-0.08 4.00
I
I
1
8.00
6.00
10.00
I
U
12.00
0.10
I
0.08
\
0.04
II 1 \ \
I
3ng ( I I 1 2
In3u In2g ( 8 2 % ) , 2og2o;:r$;(4")
3ng ( I I I )
2 0 g 20 u
1$[I!)
2 02~ 2 3i "n 3u l n2g [ 92%)
I
o,IIIl
Y&=====
2 3 3 In In ( 7 9 % ) ) ,20 2 0 In 4 In 2 [IO%) g u g g u u 3
2 2 4 2 4 2 20 In In (80%), 2 0 In l a [IO%) g u g g u g
R -+lbohr)
Figure 15. Potential energy curves of some low-lying electronic states of Te2 (reprinted from ref 236; copyright 1987 Academic Press, Inc.). See Table 37 for spectroscopic labels of known states.
2 3 3
'P+ 'P,
3P
+ 3Pz
From ref 236.
Verges et alZz9identified a singlet state (b) with a T, value of 9600 cm-'. These authors tentatively assigned the b state to '2' As seen from Table 37 the calculations of Balasu5ramanian and Ravimohan are in -0.06 agreement with this assignment. Verges et in another investigation predicted that there should be a I -0.08 'A state with an approximate T , value of 6500 cm-l, 4.00 6 00 800 10.00 12 00 altsough this state is yet to be observed. The theoR -1bohrI retical separation of the lAg state (6383 cm-') was found Figure 16. Potential energy curves of low-lying w-w states of Te2 to be in remarkable agreement with this prediction. obtained with the RCI method including spin-orbit interaction Table 37 also lists the properties of many electronic (reprinted from ref 236; copyright 1987 Academic Press, Inc.). states without spin-orbit coupling with T,values 25 700 See Table 37 for spectroscopic labels of known states. cm-', none of which, so far, have been observed exmohan2= suggested the notation A" for the 3Au(lu)state perimentally. The theoretical Re, T,, and we values for to avoid confusion with the A' state participating in the these states, however, should not be considered to be 14 091-cm-' bands. very accurate since the basis set and level of theory Ahmed and N i ~ o nobserved ~ ~ ~ another weaker employed by Balasubramanian and RavimohanB6 were fluorescence series in the 14000-cm-' region, which they not adequate to calculate these properties with reliable assigned to the forbidden A'(2,)-X(O;) transition. accuracy. However, Verges et al.235correctly argued that the A' The theoretical dissociation energy236(De)obtained state should most probably be 3Zi(lu). As seen from with the CASSCF/FOCI/RCI method for the X(0:) Table 37, the theoretical T , value of the 32:(lu) state state of Tez was found to be 1.69 eV. The spin-orbit (14 369 cm-l) is in remarkable agreement with the exinteraction decreased the De value, since the separated perimental Te value of the A' state (14091 cm-'). atoms are more stabilized by the spin-orbit interaction Further, the theoretical and experimental o, values compared to the molecule. Huber and Her~berg3~ listed agree well if the A' state is assigned to 32:(lu). Cona Doovalue of 2.68 eV based on a weighted mean of a sequently, the theoretical calculations of Balasubranumber of values obtained from spectroscopic and manian and R a v i m ~ h a supported n~~~ Verges et a 1 . l ~ ~ ~thermochemical ~ methods. The difference between this tentative assignment. value and the theoretical value is primarily due to limVerges et al.ns and Effantin et a1.230observed a B(1,) itations of theoretical calculations. As pointed out by state which was found to be close to B(0:). BalasuSaxon and L ~ u ,although '~~ Des of group IV dimers such bramanian and Ravimoha@ estimated the separation as O2 are not obtained accurately by (modest level) of this state at the Re of 32; (22414 cm-l). Although theoretical calculations, the spectroscopic properties this may not be very accurate, it is close to the expernear the well should be reasonable. imental Te of 22222 cm-'. Table 38 shows the leading configurations in the CI
iY z Q 1
1
1
126 Chemical Reviews, 1990, Vol. 90, No. 1
Balasubramanian
TABLE 39. Experimental and Theoretical Spectroscopic Constants of IZa State
T ( m-1)
ijkl
Re ( A )
Assignment
0,
Expt
("'1
Theory
Theory
Expt
2440
0
0
X
2.70
2.67
210
2431
11500
10042
A'
3.30
3.08
93
:1 9
2431
12522
11 888
A
3.30
90
93
243 1
16560
15725
a
3.25
3.32
1!C
Expt
214.5
125
6'
21000
2431
Theory
(repu! s i ve)
25700
21000
(repulsive)
(max)
2431 2341
8"
a
27900 (repu:sive)
2341
30763 (Pepulsive)
2422
a'
3170C (repu:sive)
32500
37000
(repu'sive]
;n*x:
144:
2 c22
34'30 ;-epu: 5 : ve;
35830
23Ci
('e3'"'s'je)
23cl
35369
'3-1
35233
2422
3-3r2
2332
31800
41,329
2332
41940
41521
D y
41739 2242
42500
1432
42711
3.74
Iu
5 2,
1432
(4.0)
100
3.85
3.61
41412
E('$)
-
3.67
45230
F'?
4.02
1441
47530
472!7
F
3.596
0442
5:200
47026
f
2332
53L45
H?
3.69
95 95
0'(2g]
46014
90
3.67
40396
!342
2422
3.58
90
-
105 101
94
93
102
95
3.574 1 0 3
134
3.61
101
54350 ('epJ'
5'
ie)
1432
55000
43559
r,
3.528
83
1432
56032
52030
-7
3.34
92
;342
57295
3.35
93
2332
65292
51736
F"0:
3.63
3.43
132
._I*#
i31
aTheoretical D,(Iz) = 1.45 eV including spin-orbit effects. Experimental De(12)= 1.54 eV. The notation ijkl is used to designate the electronic configuration 2a',lx',l4 A), The experimental spectroscopic c o n ~ t a n t are s ~Do0 ~ ~ ~ the state is a heavy mixture of l a ~ l a ~ 2 a 1 1 7 r ~ l 7and r~2a~ = 2.683 eV and we = 240 cm-' for the 211 (3/2,) state l a ~ l a ~ 2 a ~ l 7 r ~ 1configurations 7r~2a~ (see %able 42). This avoided crossing resulted in the maximum in the poand T, = 5180 cm-I and we = 220 cm-' for t i e 211g(1/2,) state. For the 211(3/2 ) state, the theoretical constants tential curve of the B2Z; state. At very short distance obtained by S O d / R & are De = 2.06 eV and we = 217 there is another avoided crossing. cm-', and for the IIg(1/2g) state, T, = 5979 cm-' and Table 42 shows the weights of the various configuwe = 208 cm-'. As seen from Table 40, the 21L&3/2) and rations contributing to the RCI wave functions of the 211,(1/2,) states are quite pure. Hund's rule saould electronic states of 12+.As seen from Table 42, most of the electronic states are relatively pure. The conapply in this case; i.e., the energy level of the 'II,(3/2,) tribution of 4Z;(l/2) to 211u(1/2) is noticeable (9%) as state should be lower than that of the Qg(1/2,) state. There are no experimental Re values reported for the seen from Table 42. The 4Z;(3/2) state is somewhat electronic states of Iz+. Since the (4s4p2d) basis set purer at its equilibrium geometry. yields an Re value about 0.06 A longer, Li and BalasubramanianZ5lpredicted that the experimental Re values K. TI, and TI,' of the 211,(3/2,1/2) states should both be around 2.63 A. The thallium dimer (T1,) is much more weakly bound The A state of 12+has been assigned to the zII,(l/ compared to In2 due to the inert-pair effect. For the 2,3/2) states by photoelectron ~ p e ~ t r with a ~ T ~ , , ~ ~same ~ ,reason, ~ ~ ~most of the sixth-row p-block dimers are = 12420 cm-' (211,(3/2,)) and Te = 18950 cm-' ('numuch less bound compared to the fifth-row dimers. (1/2,)). The calculated Tesand the vertical transition This will be discussed in section VII. Early experienergies from the ground state to the 211u(3/2,) and mental investigations on TlZwere focused on the de211u(1/2,) states are listed in Table 40. Table 42 shows termination of the dissociation energies. Drowart and that the 211u(3/2,) component is relatively pure while H ~ n i suggested g ~ ~ ~ an upper limit of 0.9 eV for Do0 of the Q,( 1/2,,) state exhibits considerable contamination TlZbased on failure to observe T12in T1 vapor above with the 4Z;(l/2,) component due to the spin-orbit 900 K.264 Ginter et a1.122observed emission bands in coupling, which raises the energy level of the 211,(1/2,) the 15 300-16 000-cm-' region and absorption bands in the 23000-cm-' region. It is not evident if these bands state. The photoelectron spectra of Brz revealed substanarise from Tlz or another cluster of thallium atoms. The tially different vibrational frequencies for the two related T12+ion has been observed by Berkowitz and components of the 211u state (see section III.E).37 Walter,265but no experimental spectroscopic constants Specifically, the experimental we values of the 211,(3/2) have been reported for T12. Balducci et ala2@have carried out a more recent mass spectrometric investiand zII,(l/2) states are 190 and 152 cm-'. The theogation of T1, and obtained a Do0 value of 0.63 eV using retical calculations on Brzt by BalasubramanianIs1 we = 136 cm-I for T1,. confirmed these experimental findings. The theoretical Christiansen and Pitzerlg5made small MCSCF-spiwe values of the 211,(3/2) and zII,(1/2) states of Brzt nor calculations for the electronic states of Tlz which were found to be 194 and 137 cm-l, respectively.ls1 This + zP1/zground-state T1 atoms. dissociated into the 2P1/2 anomaly was attributed to the fact that the 211,(3/2) For the 0; state, a two-configuration MCSCF treatment state of Brz is pure while the 211,(1/2) state mixes was used while for the 0; and 1, states, only one constrongly with 4Z-(l/2) and 2Z:(1/2) near the equilibrium geometry.li1 figuration was included. A double-{ STO basis set was The 211,(1/2) state of Iz+also behaves qualitatively employed. This is one of the earliest relativistic mosimilar to Brz+in that it also mixes with 4Z;( 1/ 2), which lecular calculations and thus electron correlation effects results in a lower we for the %,(1/2) component. The could not be included to high order. In a later study, Christiansenlg6carried out GVB/RCI calculations on actual amount of mixing is smaller for Iz+compared to the low-lying electronic states of Tlz. This study, Brz+. Actually, for Brz+there is an avoided crossing of
-
-
Heavy p-Block Dimers and Trimers
.os
Chemical Reviews, 1990, Vol. 90, No. 1 133
c
.04
-I
250
-
.03
In c
. --
Y
5.0 6.0
7.0
6.0
9.0
10.0 11.0
12.0 130
Bond Length (bohr)
Figure 21. Potential energy curves of two low-lying states of Tlz’ (reprinted from ref 195; copyright 1981 American Institute of Physics). TABLE 44. Molecular States of Pbz Related to Atoms in Several Low-Energy States and Their Energies” dissociation l i m i t
R (bohr)
Figure 20. Potential energy curves of nine low-lying states of Tl2 including spin-orbit effects (reprinted from ref 196; copyright 1983 American Institute of Physics). TABLE 43. Spectroscopic Constants for the Low-Lying States of T1,” State
T,
(cm-1)
R,
(a)
0
3.54
39
1,(I)
814
3.71
30
Op)
8150
3.97
25
0;
2900
3.30
a5
6200
3.41
54
6370
3.00
87
o;(IIl
6780
3.61
54
2g
8130
3.14
61
lJ!II)
9280
3.43
39
‘9
a
t
3P9
3P0
t
3P2
0.0
10 650.5
w e (cm-1)
xo;
2U
3P0
molecular s t a t e s
All values are theoretical constants from ref 196.
however, yielded only a weakly bound ground state for T12. In the earlier investigation, Christiansen and Pitzerlg5also considered the potential energy curves of the 1/2, and 1/2u states of T12+. Potential energy curves of nine low-lying electronic states of T12obtained by Christiansen196dissociating to the ground-state atoms and the excited 2P1/2+ 2P3/2 limits including spin-orbit coupling are shown in Figure 20. The theoretical spectroscopic constants are shown in Table 43. The potential energy curves of the 1/2, and 1/2u states of T12+are shown in Figure 21. As seen from Figure 20, the ground state of T12is a 0; state and is only weakly bound. The Tlz+ion in fact is a bit more bound (De = 0.58 eV) compared to T12 (Figure 20).lg5 Christiansen’slM intermediate-coupling RCI calculations gave only a weakly bound 0; state (De = 0.16 eV), in conflict with an experimental mass spectral value
“In parentheses is the number of states of a given symmetry. The energies of the dissociated atoms are in cm-’ and are from ref 134.
obtained by Balducci and Piacente266(0.6 f 0.15 eV). However, Christiansen recalculated the Do0 value by using a more accurate partition function derived from his molecular calculations as 0.37 eV. This value is in more reasonable agreement with the RCI value of 0.16 eV. Electron correlation effects are expected to be large, and thus more accurate calculations that include electron correlation effects to a higher order may bring the Do0 value of T12 closer to the revised experimental value of 0.37 eV. In any event, T12 is much more weakly bound compared to In2. Presently, there are no theoretical calculations on more excited electronic states of Ti2that would aid in the assignment of the observed spectra. Thus, such calculations are warranted to aid in the interpretation of experimental data and they are in progress.267
Shawhan268in 1935 carried out the first vibrational analysis of the observed spectrum of Pb2. Subse-
134 Chemical Reviews, 1990, Vol. 90,No. 1
Balasubramanian
TABLE 45. Spectroscopic Parameters for Pbz"
0
' 2 .O
I
I
2.5
3.0
I
3.5
4.0
1
R (A)
F i g u r e 22. Potential energy curves of the g states of Pb? (reprinted from ref 277; copyright 1982 American Chemical Society). See Table 45 for spectroscopic labels of known states.
:3 $3':
.-_
"The X, A, B, and C labels are those of Bondybey and English.87J97The theoretical constants are from ref 277.
0
2 .0
2.5
3.0
3.5
4 .0
R (A)
F i g u r e 23. Potential energy curves for the u states of Pbz (requently, many others have studied the electronic printed from ref 277; copyright 1982 American Chemical Society). spectra of Pb2 in both the gas phase and rare-gas maSee Table 45 for spectroscopic labels of known states. t r i ~ e s . ~ ~ There J ~ ~are, many ~ ~ mass ~ ~spec~ ~ * ~ ~ ~ , ~ ~ ~ ~ ~ ~ ~ trometric studies of Pb2 a1so.2631274p276 Johnson et al.272 state (also O,'), yielding a marked shoulder in the reusing the laser-induced fluorescence technique obtained pulsive side of the curve. The other component of 32;, superior spectra and showed that the early analysis of the 1, term, is lowered less by the SO effect and has a Shawhan cannot be correct. Bondybey and Englishs7," normal curve shape. Likewise the 2 curve (lA in type have studied the lead dimer LIF spectra extensively in a) is quite normal. The l2: state [without has a rare-gas matrices. second minimum at longer distances; in this region it The spectroscopic investigations of the lead dimer is primarily air: rather than 7~:. summarized above have revealed the presence of at least The 311ustate splits into 2,, l,, O,; and Of components seven low-energy electronic states. Pitzer and Balasuin increasing energy near their minima. But the 2, term bra mania^^^^^ carried out SCF/RCI calculations of ten dissociates to higher energy atoms than 1, and thus the electronic states of Pb2 and their potential energy curves cross near 3.3 A. The energy differences in these curves. u terms can be understood best by considering first the Table 44 gives the molecular states of Pb2 that dis7ri component, which yields a lower 2113/2 term if a sociate to atoms with total energies up to 22 000 cm-' high-energy a312 spinor is vacant. If a lower energy 7 ~ above the lowest states 3P0+ 3P,. Pitzer and Balasuspinor is vacant, a higher energy 2111/2term results. bramanian277have limited their calculations to those Then when the spin of the ag electron is coupled to the terms expected to be relatively low in energy (within ri group, there is a smaller splitting of 211:/2 to 2, and about 15000 cm-' of the lowest 0' state). l,, with 2, lowest, in agreement with Hund s third rule. Table 45 shows the theoreticaf spectroscopic conFrom the 2111,2term for r& there arises the 0; and 0: stants of Pb2 obtained by Pitzer and Balasubramaniterms and the second 1, term, which is 'II, in type a an277together with available experimental data. The coupling. potential energy curves are shown in Figure 22 for the The spin-orbit effect induces large changes in disg terms and in Figure 23 for the u terms, with the lowest sociation energies. The ground 0; state is only about 0; state shown in both figures. These potential energy half as strongly bound as the "Z, state without SO. curves were obtained by Pitzer and Bala~ubramanian.~~ Among the states arising from 311uon introduction of As seen from Figure 22, the spin-orbit (SO) effect the SO term, the dissociation energy of 2, is almost greatly lowers the 0; component state of 32Jair:) near unchanged whereas that for 0; is considerably reduced. Some of these differences are due to different dissocithe potential minimum. A t shorter distances this O+ state undergoes an avoided crossing with the lZi(?rt? ation limits.
Sb)
~
/
~
Heavy p-Block Dimers and Trimers
Chemical Reviews, 1990, Vol. 90, No. 1 135
-
The strongest absorption (near 19 800 cm-') is from
X(0:) to the second 0: or F state denoted by F
XSn5
close to the values obtained by Bondybev and English. Pitzerm corrected the Do0 value of Pb2 obtained from mass spectrometric measurements. This correction was obtained by recalculating the partition function of Pb2. The corrected Doo value of Pb2 was found to be in excellent agreement with the direct theoretical De value. Pacchionim2completed calculations on the low-lying states of Sn2and Pb2 after the appearance of the paper by Pitzer and B a l a ~ u b r a m a n i a n .Pacchioni202 ~~~ used the Hafner-Schwarz model potentialsa compared to the ab initio RECPs derived from DHF calculations which were used by Balasubramanian and Pitzer.m Pacchioni ignored spin-orbit effects in his computation of the potential energy curves (PEC) of Pb2 and Sn2. As demonstrated by the calculations of Pitzer and Bala~ u b r a m a n i a n the , ~ ~PEC ~ obtained for Pb2 without spin-orbit effects have little resemblence to the actual PEC which included spin-orbit effects. Consequently, Pacchioni obtained a De approximately twice the experimental value, which he subsequently corrected using a semiempirical procedure to arrive at a De close to the value obtained by Balasubramanian and P i t ~ e r . ~ ~ ~ For dimers of the sixth-row atoms, spin-orbit effects are so large that calculations without inclusion of spin-orbit terms have very little to do with the real molecule. Figure 24 shows the weights of various A-s configurations in the D states of Pb2. As seen from this figure, for Pb2 the A-s populations change dramatically as a function of internuclear distances. At short distances, there is an avoided crossing of ?r;('Z+) with ~i7ri(~Z;). This results in the shoulder of the X[Oi) curve of Pb2. At long distances many configurations contribute, leading to dissociation into 3P0+ 3P0atoms instead of
An energy curve for this state was added by Pitzer and B a l a ~ u b r a m a n i a nto~ Figure ~~ 23 at the experimental energy. Since this state arises primarily from the configuration cr2?ru?rg, it is expected to have a larger Re than that for C T ; this ~ was found to be in agreement with the red shading of the experimental bands. The transition probability from 0: to the lower 0: state was found to be experimentally much smaller, in agreement with the nature of these states. Theoretical spectroscopic properties obtained by Pitzer and Balasubramanian for this state agreed reasonably well with those measured, in particular the large anharmonicity and low dissociation energy. Also the theoretical Re value for the lowest 0: is smaller than that for ,O: in agreement with the blue shading of the observed Bondybey and E n g l i ~ h ~interpreted ~J~~ a set of emission bands for Pb2 in an inert matrix as arising by internal conversion from the F state to the long-lived B state at 12 457 cm-l. Although this state is not connected to the ground state by a dipole-allowed transition, it was found to radiate slowly as a result of matrix distortion or higher order effects. Bondybey and Englishlg7suggested the 2, state, but its energy is much too low as seen from Table 45. However, theoretical calculations of Pitzer and Bala~ubramanian~~~ indicated this to be the 0; state. Matrix spectra exhibited emission bands near 13400 cm-' connecting two new states. Since the upper D state arises by internal conversion from the F state, the lower or A state cannot be more than 6540 cm-' above the ground state. Pitzer and Balasubramanian argued that the most probable assignment for the A state is 1,. 3P + 3P. The theoretical Teof the 1, state was found to be 4150 cm-l. The 2, state cannot be completely eliminated from consideration since the Pitze~Balasubramanian~~~ M. Bi, theoretical energy of 6670 cm-l might be in error by several hundred cm-'. The theoretical we of the 1, term There are many experimental spectroscopic investiwas found to agree quite well with the experimental g a t i o n ~ ~on ~ *Biz ~ ~and ~ Bi4 clusters. In fact, for value. The D state at an energy of 17500 cm-' (or 19800 sometime a band that was actually due to Bi4 was incm-' if the A state is 2,) could not be calculated with correctly attributed to Bi2. This interpretation led to reliable accuracy.277 another new ground state for Bi2. This confusion has now been resolved, and since then the ground state of The theoretical De for the 0: ground state of Pb2 of 0.88 eV277was almost in exact agreement with an exBiz has been unambiguously established as a 0: state. perimental value of 0.86 f 0.01 eV obtained by C h r i s t i a n ~ e ncarried ~ ~ ~ out generalized valence bond Gingerich et al.276 Although this agreement was concalculations followed by RCI calculations. A double-{ sidered fortuitous by Pitzer and B a l a ~ u b r a m a n i a n , ~ ~ ~STO basis set was employed in the GVB calculations. Figure 25 gives the potential energy curves of Bi2under the agreement of such magnitude on a very heavy dimer various approximations. The spectroscopic constants such as Pb2, for which both spin-orbit effects and reof two well-characterized states of Biz are shown in lativistic effects are large, must be considered impressive. Table 46. Among the curves in Figure 25, the curve Sontag and Weber278obtained the Re value of the labeled FV7R (limited full valence + seven reference ground state of Pb2 from experimental spectra after the singles + doubles CI) is the best a t this level of a publication of the theoretical calculations on Pb2 by proximation. These calculations yielded Re = 2.79 , Pitzer and B a l a s ~ b r a m a n i a n .The ~ ~ ~experimental Re we = 170 cm-', wexe = 0.3 cm-l, and De = 2.3 eV.= The of 2.930 8, agrees very well with the earlier theoretical corresponding experimental values for the X(0:) state value of 2.97 A.277 In general, Re values obtained by are Re = 2.66 A, we = 173 cm-', wexe = 0.34 cm-', and RECP/RCI calculations tend to be a bit longer, and De = 2.04 eV.37 The theoretical Re is expected to be a thus this agreement should be considered excellent. bit longer at this level of approximation, and thus the The blue-green system that Bondybey and English87 0.13 8, longer bond length for Bi2 is not surprising. called the F X system was studied further by Berg However, as seen in section III.H, for Sb2 a more soet in laser-induced fluorescence. From these phisticated and accurate CASSCF/MRSDCI/RCI spectra the spectroscopic constants of the X and F calculation yielded Re = 2.59 8, compared to an exper(Berg et al.279call the F state B) states were obtained imental value for the X(0:) ground state of 2.48 f 0.02 by these authors.279The constants thus obtained were A. Consequently, even calculations that included
x
-
136 Chemical Reviews, 1990, Vol. 90,No. 1
Balasubramanian
TABLE 46. Experimental Spectroscopic Constants for Bizn
‘,Or-----
State
xo; 3 2,o; -
3,
(AI
(cm-’)
T,
LJ
(“:I
2.66 (2.79)
0
173 ( 1 7 0 )
2,863
17 739
132.4
Experimental D,(Bi,) = 2.04 eV compared to a GVB/RCI value of 2.3 eV.292 The state labeled B’ was also called A by Ehret and Gerber.=l These authors obtained the transition moment of the A-X system as 1.4 f 0.4 D. The lifetimes of the vibrational levels (u’ = 1-34) ranged between 50 and 600 ns. Theoretical values are in parentheses and are from ref 292.
Rlbohr
Figure 24. Fractional A-S populations with low-lying states of Pbz and Snz (reprinted from ref 201; copyright 1983 American Institute of Physics).
effects and to a smaller extent due to the inert-pair effect. CASSCF/FOCI/RCI calculations on both the ground state and excited states of Biz are being made in the present author’s group. Such calculations should especially be of use in comprehending the electronic spectra of Bi2 and the nature of the excited electronic states of Biz.
I V. Spectroscopic Properties and Potentia/ Energy Curves of Heteronuclear Dimers A. GaAs and GaAs’
4s
50
5.5
R
so
65
7.0
75
(bohr)
Figure 25. Potential energy curves for the ground state of Bi, using various methods (reprinted from ref 292; copyright 1984 Elsevier Science Publishers B.V.).
electron correlation effects to a much higher order yielded slightly longer bond lengths within the RECP approximation. The theoretical RCI Devalue of Biz (2.3 eV) obtained by C h r i ~ t i a n s e nwas ~ ~ ~surprisingly larger than the corresponding experimental value of 2.04 eV. This primarily appears to be due to size inconsistency in the treatment of spin-orbit coupling in small-scale RCI calculations. Analogous to Pb2,the De of Bi2 should be much larger when the spin-orbit coupling is omitted. For Pb2, note that the Dewithout spin-orbit coupling is twice the true value (see section 1II.L). Spin-orbit coupling thus stabilizes the atoms more than the molecule destabilizing the bond. Consequently, the higher Deobtained by C h r i ~ t i a n s e nshould ~ ~ ~ be attributed to size inconsistency at long distance in the treatment of the spin-orbit term. The spin-orbit contamination of the lZi(0;) state with the 311g(Oa) state is of interest since this contamination destabilizes the bond. The X(0:) state of Bi, was found to have 25% 311gcharacter. Since the 311g state arises from occupying the antibonding rgorbital, this would lead to a bond order lower than 3.0 for Bi2. Christiansenm approximately calculated the bond order from predominant configurations as 2.17. Evidently, the bonding is much weaker both due to spin-orbit
GaAs semiconductors are of immense technological value since they are useful in constructing fast devices among other applications. As a result, a number of investigators have studied the properties of GaAs surface~.~~~-~@’ Smalley and.co-workersg7have generated supersonic beams of semiconductor clusters of formula Ga,Asy by laser vaporization of pure GaAs. Cluster beams were characterized by laser spectrometry. Mass analysis of the large clusters revealed a binomial distribution while the abundance of smaller clusters deviated considerably from the binomial distribution. In particular, the diatomic clusters were found to be mostly GaAs and As2 with very little Ga2. Further, photoionization with an ArF excimer laser revealed even-odd alternation in the photoionization cross sections. Balasubramanian301 made complete active space MCSCF calculations followed by first-order CI calculations of potential energy curves and spectroscopic properties of the low-lying electronic states of GaAs and GAS+. He considered 12 electronic states of GaAs and 4 electronic states of GaAs+. Valence Gaussian (3s3pld) basis sets for Ga and As atoms in conjunction with RECPs that retained the outer 4s24p1and 4s24p3shells of Ga and As, respectively, were used in these calculations. After these calculations were published, M. Morse and co-workers at Utah completed their experimental spectroscopic study on GaAs. The bond lengths calculated by Balasubramanian were longer than the experimental values of Morse. This motivated Balasubramanian to reinvestigate GaAs. An error was found in one of the ECP parameters, which was subsequently corrected. The corrected spectroscopic constants for GaAs were, however, found to be in good agreement with both experiment and all-electron calculations. Following the theoretical calculations of Balasubramanian,301 Knight and Petty302 observed the ESR spectrum of GaAs+ by laser-evaporated GaAs followed
Heavy pBlock Dimers and Trimers
Chemical Reviews, 1990, Vol. 90, No. 1 137
by photoionization. The experimental spectrum confirmed the 42- ground state for GaAs+ predicted by Balasubramanian. Lemire et al.303have recently studied the optical spectra of jet-cooled GaAs generated by laser vaporization of a GaAs crystal. These authors observed bands in the 23 000-24 000-cm-' region. All the bands except the 311(O+)-X32-(O+)bands were found to be predissociated above the u = 1 level. Basically, the results of these calculations confirmed Balasubramanian's prediction301of a 32- ground state for GaAs. The possible low-lying electronic states of GaAs and GaAs+ are shown in Tables 47 and 48, respectively. Three possible candidates for the ground state of GaAs are 32-,311, and l2+,while the corresponding candidates for the ground state of GaAs' are 42- and Q. The CASSCF/FOCI calculations of Bala~ubramanian~~l included up to 592 configurations at the CASSCF level and 26 200 configurations at the FOCI level. Tables 49,49A, and 50 show the spectroscopic properties of GaAs and GaAs+,respectively. The theoretical potential energy curves of some of the electronic states of GaAs and GaAs+ are shown in Figures 26 and 27, respectively. The ground states of GaAs and GaAs+ are 32-and 42-,respectively. As seen from Table 49, there are six bound states for GaAs below 18000 cm-l. None of these states except the X ground state and 311(111) state appear to have been characterized experimentally. Some of the allowed electric dipole transitions among the states in Table 49 are 311 32-,'Z+(II) l2+,and l2+ 'II. These transitions are observable in the neighborhood of 1500-1600, 8200, and 1315 cm-l, respectively. Table 51 shows the vertical and adiabatic transition energies of several allowed electric dipole transitions. As seen from that table and Figure 26, the 32-(11)and 3Z-(III) states are not bound electronic states. Thus, transitions from the ground state to those excited electronic states could be observed only as broad absorption bands. These transitions occur (Table 51) in the region of 20000-30000 and 35000 cm-l, respectively. Hence Balasubramanian301 predicted two broad absorption bands in this region. The 42-(11) 42-transition of GaAs+ was predicted to lie in the region of 40 000-42 000 cm-l. The '%(II) zII transition was predicted to be observable in the 9186-cm-l region. The vertical ionization energy of GaAs was calculated as 6.85 eV. The ionization potential of the As atom was calculated from asymptotic splittings as 9.76 eV. The corresponding experimental value134is 9.81 eV. Consequently, the agreement is good. The theoretical FOCI dissociation energy of GaAs is 1.25 eV. The GaAd ion is much less stable compared to neutral GaAs, since its De was found to be only 0.36 eV. The theoretical dissociation energies of Gaz and Asz were found to be 1.2 and 2.71 eV, respectively (see sections III.A,C). The calculated Des of these species were found to be consistent with the observation of O'Brien et aLg7that the diatomic species detected upon laser vaporization of the GaAs crystal were mostly Asz and GaAs with very little Gaz. The CASSCF/FOCI calculations in general yield a De in the range of 75-85% of the experimental values. Thus, the De of GaAs should be higher than the theoretical value. After publication of Balasubramanian's theoretical
-
-
-
-
TABLE 47. A Few Low-Lying Molecular States of GaAs Arising from Atomic States of Ga and As Molecular s t a t e s
32-, 37, 52-, 511
Atomic s t a t e s
Ga
+
As
2P
t
4s
Energy o f t h e s e p a r a t e d atomsa ("1)
0
TABLE 48. A Few Low-Lying States of GaAs+ and Their Dissociation Limits Molecular s t a t e s
4,2 A , 211,
3-
Atomlc s t a t e s
Energy o f t h e s e p a r a t e d atomsa ("1)
Gat(lS)
t
As(~S)
0.0
Ga*(lS)
+
As('0)
10 790
-
calculations,3°1there were two experimental studies on GaAs and GaAs+ as mentioned before. Knight and Petty3OZrecorded the ESR spectra of GaAs+ generated by laser evaporation of a GaAs crystal followed by photoionization. The ESR spectra revealed a quartet structure, confirming Balasubramanian's prediction of a 42-ground state of GaAs+. Lemire et al.303studied the rotationally resolved electronic spectra of jet-cooled GaAs diatomic. These authors obtained a 32-ground state for GaAs, confirming Bala~ubramanian's~~l prediction of a %- ground state. The experimental spectroscopic constants obtained from the analyses of the observed bands by Lemire et al.303are shown in Table 49A together with Balasubramanian's theoretical constants."' As seen from Table 49A, theoretical constants obtained by Balasubramanian are in very good agreement with experimental values. The ground state of GaAs ( 3 2 - )was found to be predominantly (89%) la22a23a217r2(3Z-) at its equilib-
138 Chemical Reviews, 1990, Vol. 90, No. 1
Balasubramanian
TABLE 49. FOCI Spectroscopic Constants for GaAs" State
R,
(1)
T,
(crr-'~
1F
3, [ ? I
jcn-::
TABLE 50. Spectroscopic Constants for GaAs+a Re (A)
State
T,
FOCI
SOC I
2.877
2.944
(cm -l)
we (cm
SM: I
FOCI 0
FOCI 0
103
2.519
9 707
250
2.968
11 604
134
2.632
2.719
13 034
12 148
18 600
2.560
141
-'I SOCI
89
125
293
!52 18 155
177
2.530
:E.:
2.624
30 024
168
2.907
33 854
100
143
2.653
20 551
2::
138
2c1
(I
2.592
34 070
33 279
218
:69
2.566
2s;
2.938
36 237
126
.. c"/-
2.346
39 982
362
2.909
46 290
40 2
210
All constants are from ref 301 and 404.
TABLE 49A. SOCI Spectroscopic Constants and Known Experimental Constants for GaAsa State
R,
(A)
T, (cm-lj
we
Theory
Expt.
Theory
X35-
2.60b
2.516
0
3l
?.38
1833
236
-Et
2.23
7758
279
1~
2.58
7574
ZlC
2.47
id383
321
:a590
135
:z+!II! 'S(!I:\
2.68
2.66
24600
Expt.
3
23830
(cm-')
Theory
Expt.
215
214
153
132
Experimental values are from ref 303; theoretical values are from ref 404. *An Re value of -2.55 A is estimated if d correlation effects are included. (I
rium geometry. The 32-state arising from 1$2217r227r2 makes a small contribution (2.4%). A t long distances the contributions from la22023a22a2as well as lo22a23a4a17r2increase. The lZ+state of GaAs exhibited an avoided crossing reminiscent of the corresponding state of the isoelectronic Gez. This state is predominantly lo22a21a4( lZ+) at short distances and near equilibrium geometries. At its equilibrium geometry, the lZ+ state is 73% and 12% lo22a23u217r2('Z+). The 1a22a23u21a2configuration dominates at longer distances. Thus, this state resembles the lZ+ state of many group IV dimers such as Ge2,139Sn2,201and Pb2.277 The 313 and l l 3 states of GaAs are predominantly la22a23ala3(87%). As seen from Figure 26, the 3Z-(II) and 3Z-(III) curves are repulsive. A number of configurations such as 1022a23a217r2a,1a2203a1x32a,and la22a21n32ncontribute to these states. The 'Z+(II) state is 52% 1a22a23a21s2, 16% la22a217r4,and 10% lu22a3als4at its equilibrium geometry. The l A state
All constants are from ref 301 and 404.
was found to arise dominantly from the la22u23a21x2 Configuration. The 42-ground state of GaAs+ is primarily composed of the la22a23a17r2(92%) configuration at its equilibrium geometry. The contribution of this configuration does not change much as a function of internuclear state of GaAs+ exhibits an indistance. The 4Z-(II) teresting behavior as a function of internuclear distance. The contributions of various configurations to the 4Z-(II) state for a few distances are shown in Table 52. As seen from Table 52, this state is predominantly la22a13a21a2at short distances, although other configurations such as la22a23ala2amake a significant contribution. As the distance increases, the contribution of la22a23a17r2aincreases accompanied by a decrease in the contribution of la22a13u217r2configuration. At distances >2.75 8,these two configurations make equal contributions, indicating an avoided crossing. At 3.00 A, the contribution from la22a23a17r27r(49%) dominates. The contribution from this configuration decreases as R increases further and the contribution from 1a22$4a17r2configuration increases. At longer distances the la22a24a17r2configuration dominates. The two avoided crossings in this state explain the shape of the 4Z-(II) curve (Figure 27). A t long distances the l o , 2a, 30, and 4a orbitals are dominantly A&), Gab), As(p), and Ga(p), respectively, whereas the l a orbital was found to be predominantly As(p). Thus, the la22a24u17r2configuration corresponds to Ga(s2p1)and As+(s2p2)or Ga?P) + As+(~P).This explains the domination of the 1a22a24u17r2configuration at long distances since the 4Z-(II) state dissociates into GaPP) + As+(~P). At short distances the 211state was found to be primarily composed of the la22a217r3 configuration (82?'0).~~However, at 2.50 A, it is made up of l$2a217r3 (54%) and la22a23a21a(32%). For R > 2.50 A, this state was found to be a mixture of la22a217r3and
Heavy p-Block Dimers and Trimers o.20
0 15
-
Chemical Reviews, 1990, Vol. 90, No. 1 139
TABLE 51. Vertical and Adiabatic Energies of the Allowed Electric Dipole Transitions of GaAs"
1
I'
Go As
Transition
1
Te
32- * 3n
1 830
--
3 ~ - 3 ~ - ( ~ ~ )
__ --
2 9 730
+
In
0.10
0,
E
3 ~ -32-(~~~)
-
+
3n
e L O
I
3~
t
-
34 548 31 220
* 35 +
*
0.05
Tvert (cm-'~
32-(11)
--
11+
1 249
1 371
10 007
9 969
32 946
W
* lZ+(II)
a
11' * lZ+(II) 0- I
"From ref 301. TABLE 52. Contributions of Leading Configurations to the %(II) State of GaAs+"
-
-0.05
I
2.00
4.00
3.00
R
5.00
I
0"'"1
Contrih u t i o n
R(A)
6.00
--)(A)
Figure 26. Potential energy curves of some low-lying states of GaAs (reprinted from ref 404; copyright 1990 Academic Press, Inc.).
-
io 298
758
2.50
la 22a 1302 In2 ( 5 2 ) , la22a23aln2n ( Z O ) , 2 1 2 2 2 la 2a3a21n2n (6),loZ2a24aln2( 5 ) , la 20 30 In ( 3 )
2.75
GaAs'
lo22a3a21nZ (34), lo22a23aln2n (26), la2203021n2n ( l o ) , la22a2401n2 (6)
3.00
0.10-
al
laZ2a23aln2n (49), 1 0 ~ 2 a 3 3 ~ 1(111, n~ la22a3a21n2n (11), l a 2 2 a 2 3alnnRy (10)
e
c h
O
I
T
3.50
la22a23aln2n (39), la22a24al~2(26), lo2202331nn ( l o ) , l a 2 2 0 2 a in 2 ( 5 ) RY RY
3.75
la 2202401 I n 2 (561, la 2 2a 2 3 in 2 (13),
0'05-
W
RY
"
*
I
2.00
300
4.00
R
5.00
6.00
4.00
+(A)
Figure 27. Potential energy curves of four electronic states of GaAs+ (reprinted from ref 301; copyright 1987 American Institute of Physics).
la22a23$1?rconfigurations. The 211(11)state was found to be predominantly la22a23a21?r(87%) at short distances. A t 2.50 A, it was found to be a mixture of la22a23a21?r(57%),la22a217r3(26%),and la22a21?r22a (4%) configurations. At 2.74 A, the 211(11)state is composed of la22a21a3(60%),la22a23a21x(13%), and la22a21?r22?r (5%) configurations. Thus, in this region this state exhibited an avoided crossing. At longer distances the lu%Pl?r3 configuration dominated so that this state would dissociate into Ga+('S) + As(~P). After the appearance of Balasubramanian's paper, Meier et al.403published an all-electron study of the electronic states of GaAs. They also found a 32-ground state for GaAs and a very low lying 311 state (T, 0.17-0.24 eV) above the ground state using the Hartree-Fock/MRDCI method. Most of the discrepancies between Balasubramanian's earlier resultsm1and Meier et al.'s all-electron results403are due to an error in one of the ECP parameters mentioned before. The bond
-
" Ry suffix indicates that the orbital is a Rydberg orbital from ref 301. Numbers in parentheses are contributions in percentage. lengths in Table 49 are within 0.04-0.05 A of the allelectron results for these states. The revised wes are also in much better agreement with the all-electron results. It is, however, worth noting that the l A state has a larger we if 'Al CASSCF orbitals are used in the calculations while the 'A2 orbitals yield a more reasonable w,. Hence the larger we for the 'A state obtained before301is due to the change of 'Al orbitals due to avoided crossing of the la22a21a4and la22a23a2?r2configurations in the CASSCF calculations. Meier et al.* obtained Re = 2.60 A and we = 202 cm-l for the X32- ground state of GaAs using the all-electron MRDCI method. Balasubramanian's calculations included electron correlations to a much higher order compared to a restricted configuration space of less than 10000 configurations in the MRDCI. Although the FOCI method includes most of the significant electron correlations, a full SOCI method is much superior to
140 Chemical Reviews, 1990, Vol. 90, No. 1
t
t
Br'f'D!
Balasubramanian + Kr
1's)
L A B ENERGY lei/
Figure 29. Energy spectrum of subelastic and superelastic collisions of Brt with Kr a t a relative kinetic energy of 4 eV (reprinted from ref 326; copyright 1986 Elsevier Science Publishers B.V.). Figure 28. Energy level diagram of the Brt-Kr system at infinite interatomic separation. T h e observed transitions are indicated (reprinted from ref 326; copyright 1986 Elsevier Science Publishers B.V.).
both the MRDCI and FOCI methods. Even at the FOCI level Re = 2.645 A and we = 187 cm-l of the 32state are in good agreement with the all-electron MRDCI calculations. It is worth noting that although Meier et al. employ an all-electron method, they do not include relativistic effects while Balasubramanian's calculations took into account relativistic effects. Meier et al. obtained Re = 2.38 A and we = 260 cm-l for the 311 state while Balasubramanian obtained Re = 2.37 A and we = 241 cm-l at the FOCI level. Similarly, good agreement was found for the other states listed in Table 49 with the exception of the 'Z+(II) state, for which FOCI calculations starting from a CASSCF orbital set of 'Al symmetry yield a larger we of 420 cm-l. However, when SOCI calculations are made using FOCI natural orbitals, this discrepancy vanishes and thus the higher we of the 'Z+(II) state at the FOCI level must be concluded as due to the strong variation of the composition of the CASSCF orbitals due to the avoided crossing of the lo22a217r4and la22a23a217r2configurations. The D,(GaAs) obtained at the FOCI level is 1.24 eV compared to a MRDCI value of 1.40 eV. However, SOCI calculations when corrected for d correlation effects yield a much better De of 1.9 eV in closer agreement with the value predicted by Lemire et al. For GaAs+, the De obtained by Meier et al.403is only 0.11 eV compared to Balasubramanian's De of 0.36 eV obtained with the full SOCI method. Thus, the MRDCI method yields much poorer Des compared to a full second-order CI method due to truncation of configurations in the MRDCI method. However, Re and we values obtained by the MRDCI method were found to be in good agreement with Balasubramanian's SOCI results, which he obtained in a more recent ambitious CASSCF/SOCI study404on GaAs, GaAs-, and GaAs+ that included up to 700 000 configurations. Balasubramanian305 has recently completed CASSCF/FOCI/MRSDCI calculations on many lowlying electronic states of the iso-valence-electronic diatomic InSb. Fifteen low-lying bound electronic states (Te< 26 000 cm-') were found for InSb without spin-orbit effects. When spin-orbit coupling was in-
cluded, more electronic states resulted. Analogous to the diatomic GaAs, the ground state of InSb dimer was found to be a 32- state. The 311 state of InSb was considerably higher (T, 2300 cm-l). The Re and we values of the X3Z-,+ ground state of InSb were found to be 3.0 A and 121 cm-l, respectively. The adiabatic IP and electron affinity of InSb were calculated as 6.33 and 1.41 eV compared to 6.85 and 1.3 eV for GaAs.
-
B. KrBr' and Energy Transfer in Br+-Kr Collislons
The interest in rare gas-halogen neutral and ionic compounds arises partly because these are potential candidates for chemical l a ~ e r s . ~ The ~ - ~generation ~l and deexcitation of neutral RgX species have also been extensively studied. Rare-gas oxides have also been the topics of many investigations since they are potential candidates for optical pumping reactions. In the collision of halogen positive ions with rare-gas atoms there is an efficient transfer of electronic energy to translational energy and vice An energy level diagram of the Kr-Br+ system at infinite interatomic separation is shown in Figure 28. Since the excitation experiments by Koski and c o - w ~ r k e rwere s~~ carried out at relative energies of
3 2
3
c
c
-0.962
!al
?al
3a:
2
2
i
2 3 2
4.
"1
-0.951 C.113
:a1
la1
32:
4ai
:52
2
3 3
2 2
1 2
3 0 I 2
2 2 2 :
2 ; 1
2
2
:
0 5
!a,
23:
32:
2 2
2 1
0
2:*
3,
-0.940 -3.i27 -c.i:4 c.:32
2
:
2
2
0
0 2
l
0 0
0 0
2 :
: 3
3
c '2
0
3
1 :
0
"From ref 355.
structure which is corrected to a near-equilateral triangle structure by POLCI. The energy separations are, in general, more sensitive to electron correlation effects. The CASSCF atomization energy was calculated by carrying out a long-distance CASSCF calculation for the linear ,A1 geometry ( R = 8.0 A). The atomization energy calculated in this way was found to be 2.32 eV.355 Higher order correlation effects not included in the CASSCF can certainly increase the atomization energy. The corresponding CASSCF De of the Gaz dimer calculated earlier by Bala~ubramanianl~~ is 0.92 eV. Consequently, Ga, is at least 1.4 eV more stable than the dimer. The corresponding calculations on Al, by B a ~ c h "revealed ~ that the aluminum trimer is about 1.93 eV more stable than the dimer while the experimental results of Fu et al.358imply Do(Alz-A1) < 2.4 eV. Table 63 depicts the dipole moments of the isosceles triangular states of Ga, calculated from the POLCI natural orbitals. As seen from Table 63, the ground state ?A1) and ,B1 excited states have nearly vanishing dipole moments while the other electronic states have nonnegligible dipole moments. The positive charge is on the apex atom of the isosceles triangle. The 4Az,4Bl, 2B2,and 4B2states of Gas are slightly ionic while the 2A, and 2B, states are essentially nonionic. As seen from Table 62, there are two low-lying electronic states for the Ga3+positive ion. The ground state is a closed-shell 'A, state with a linear geometry. The adiabatic ionization energy of Ga3 is 5.99 eV. The 'A1-3Bz splitting of the Ga3+positive ion is 0.65 and 0.37 eV at the CASSCF and POLCI levels of theory, respectively. Consequently, this energy separation is influenced to a large extent by higher order correlations. Most of the ionization occurs at the base atoms of the
triangle than the apex atom as evidenced from the Mulliken population analysis. The adiabatic ionization potential of the Ga3 cluster (5.99 eV) is quite comparable to the experimental ionization potential of the Ga atom,134which is 6.0 eV. It is interesting to note that both the Ga atom and Ga3 clusters form a stable closed-shell singlet electronic state upon ionization leading to lower adiabatic ionization energies compared to even clusters. Hence odd-even alternations are anticipated in ionization energies. The nature of the electronic states of Ga, is discussed next. The lowest la, orbital is the totally symmetric bonding combination of the 4s orbitals of the three Ga atoms. The 2al orbital is also predominantly composed of the 4s orbitals on the three Ga atoms with the combination -4s(l) + 4s(2) + 4s(3), where the label 1 is given for the central apex atom. Consequently, the 2al orbital is antibonding along the sides and bonding along the base of the triangle. The l b porbital was predominantly found to be the 4py(l) + 4s(2) - 4943) orbital. Hence this orbital is antibonding along the base and bonding along the sides. Consequently, the lb, orbital is stabilized if the sides are shorter than the base of the triangle. Conversely, the 2al orbital is stabilized if the sides are longer than the base of the triangle. The lb, orbital is a symmetric combination of the 4p, orbitals on the three Ga atoms perpendicular to the plane of the cluster and thus is "r-like" in character. The 3al orbital is a bonding orbital resulting from the mixing of the 4p, orbital of the central atom with the 4py and 4p, orbitals of the base of the triangle. The 4s orbitals on the three atoms also participate in this orbital. Similarly, the 2b2 orbital is a bonding orbital resulting from the interaction of the 4p, orbital of the central atom with the
Chemical Reviews, 1990, Vol. 90, No. 1 149
Heavy p-Block Dimers and Trimers
Ga3 2
2.51
5.52
1..657
3.291
0.782
1.980
0.026
0.067
2.993
6.007
1.647
3.299
1.232
2.479
0.114
0.230
0.959
2.634
5.435
1.667
3.515
0,919
1.717
0.027
0.065
3.100
5.900
1.629
3.429
1.361
2.232
0,109
0.238
0.931
2.609
5.612
1.780
3.662
0.814
2.002
0.037
0.097
2.998
6.002
1.703
3.416
1.184
2.353
0.111
0.233
0.778
B2
2.550
5.442
1.547
3.166
0.892
1.728
0.024
0.052
3.054
5.946
1.588
3.311
1.353
2.393
0.113
0.242
1.008
51
2.602
5.842
1.793
3.392
0.726
2.220
0.024
0.072
2.880
6.120
1.738
3.351
1.024
2.529
0.117
0.240
0.556
B2
2.774
5.813
1.888
3.623
0.847
1.989
0.020
0.066
2.981
6.019
1.799
3.533
1.096
2.263
0.085
0.223
0.413
A1
2.665
5.831
1.695
3.906
0.849
1.861
0.023
0.051
2.917
6.083
1.691
3.619
1.097
2.320
0.129
0.144
0.504
A1
4 A2 2 2 4 4 4
Ga; 1 A1
3
2.819
4.990
1.673
3.931
0.999
0.829
0.021
0.065
2.915
5.085
1.708
3.555
1.073
1.262
0.134
0.268
0.190
2.857
4.736
1.555
3.801
1.127
0.675
0.024
0.048
3.060
4.940
1.624
3.560
1.300
1.122
0.137
0.258
0.406
OM2refers to the two equivalent atoms of the cluster. From ref 355. *Total overlap of the central atom with the two side atoms.
antisymmetric combination of the 4p, and 4p, orbitals of the base atoms with appropriate signs. The 2bl, 4a1, and la2 orbitals are antibonding orbitals. Table 64 contains the important configurationws contributing to the CASSCF-CI wave functions of the electronic states of Ga, and Ga3+. As seen from Table 64, the leading configurations have coefficients 10.9 for all the electronic states. The second and third leading configurations have coefficients >0.1 for the doublet states. The second leading configuration is less important for the quartet states with the exception of the 4A1 state. In summary, the 4B2and ,B2 states of Ga3 and Ga3+ion are described well by a single-configuration treatment. The other electron states seem to require a multiconfiguration treatment. The 'Al state of the Ga3+ion is predominantly lai2a:lbi2bi, suggesting that this state is formed by removal of the 3al electron from the 2Al ground state of the neutral cluster. This is anticipated since the single electron resides in the 3al orbital of the 2Al state of Ga,. Table 65 depicts the Mulliken populations of the POLCI natural orbitals of the various electronic states of Ga, and Ga3+. The total s populations of the three metal atoms in all the electronic states are between 4.90 and 5.33. The 2B2and 2A1states have the smallest s populations while the 4B2and 4A1states have the largest s populations. The corresponding p populations are between 3.36 and 3.75. The 2Al and 2B2states have the largest p populations while the 4B2and 4A, states have the smallest p populations. Hence there is considerable 4s to 4p electron transfer in the electronic states of Ga,. The d populations are about 0.3-0.35 for all the electronic states of Ga3. Consequently, the participation of the d polarization functions is quite important for all the electronic states of Gas. A comparison of the total gross populations of the central and side atoms
of Ga3 and Ga3+ (2A1, lAl) revealed that most of the electron removed in the ionization process comes from the base atoms than the central atom. Of course, there is considerable rearrangement of this geometry from a near-equilateral-triangular to a linear structure upon ionization. Comparison of the total s, p, and d populations of the ground states of Ga3and Ga3+shows that the electron is removed from the 4p orbital and that there is considerable 4p 4s electron transfer upon ionization. The electronic states of Al, show considerable similarities to the corresponding states of Ga,. B a s ~ h ~ ~ ~ found the ground state of A13 to be a 2A1state with an isosceles triangular geometry (apex angle = 56.2'). The 2Bl and 4A2 states were found to be 0.22 and 0.23 eV above the ground state. The energy separations for Al, are thus comparable to the corresponding electronic states of Gag. The geometries are, however, somewhat different as expected. The A13+ion has a 'Al closedshell ground state. The 'A1-3B2 splitting for A13+was found to be 0.42 and 0.39 eV at restricted and full levels of CASSCF calculations, respectively. The POLCI 'A1-3B2 splitting for Ga,+ obtained by Balasubramanian and Feng355(0.37 eV) is quite comparable to the corresponding splitting for A13+.
-
6. GaAs,
Presently, there are no experimental spectroscopic parameters of GaAs2, although it has been observed among trimers by Smalley and co-workersg7in a laserevaporated jet beam of general composition Ga,As,,. B a l a ~ u b r a m a n i a ncarried ~ ~ ~ out CASSCF/multireference singles + doubles CI calculations on the low-lying electronic states of GaAs2. Three lowest lying states were found for GaAsp. All these calculations were done
150 Chemical Reviews, 1990, Vol. 90, No. 1
Balasubramanian
TABLE 66. Geometries and Energy Separations of GaAs," Re ( A !
State
Be
E (eV)
( 0 1
TABLE 67. CI Wave Functions of the 2B, and 2A,States of GaAs, at Their Equilibrium Geometriesa 2
2
B:
2.76
/A!
2.37
58.5
0.67
%
2.75
51
1.7
47
0
"MRSDCI geometries and energies for the 2B2and 2A, states. CASSCF geometry for the 2Bl state. Energy of the 2B1state based on the CASSCF 2B,-2Al separation. Re refers to the Ga-As equal bond lengths while 8, is the As-Ga-As bond angle. From ref 359.
A5 2 201
'B2
A
AS
0 kcoi/mole
a s - ~ - A s 2 326
'Al
16 kcal/mole
as-
c3e f f
7,
0.919
-.! 0 6
c 1 en t
82
configuration lal
2al
3al
Sa1
Sal
2
2
2
2
3 . 2
2
2
2
2
0
2
2
2
l
2
2
2
1
131
la7
1
2
0
2
1
0
2
C
2
2
2
0
0
2
2
0
2
Ib2
252
As 2 37;
' 8 , 39kcQl"ole
Figure 36. Equilibrium geometries and energy separations of the low-lying electronic states of GaAs,.
by using RECPs for the various atoms together with valence (3s3pld) basis sets for Ga and As. The CASSCF calculations included up to 3600 configurations while the MRSDCI calculations included up to 93 000 configurations. In all the theoretical calculations, the GaAs2molecule was oriented on the yz plane, and the z axis bisected the As-Ga-As angle. The x axis was perpendicular to the GaAs, molecular plane. If one assumes the GaAs, structure to be triangular, a few possible low-lying doublet states arising from the outer s2p' and s2p3shells of Ga and As in the CZugroup in this orientation are laT2a:lbi3aflbf4af2b; (2B2), lay2atlbi3aflbf2ai4a: (2A1), laf2ailbi3af4af2b;lb: (2B1), and l a : 2 a f l b ~ 3 a ~ l b f 4 a f(2A2). l a ~ While quartet states could be generated by promoting one of the closed-shell electrons into an unoccupied orbital, they would be much higher since the two electrons of the As atoms tend to pair up to form a stable configuration similar to the ground state of As2. If the ground state of As, were to bind to the Ga ground state (2P)it would form doublet states. Further, attempts by Balasubramanian359to find an equilibrium structure for the 4B2state failed. The 2A2 state is also expected to be higher in energy since this state results from occupying the antibonding combination of the px orbitals of the As atoms. The states resulting from correlating the ground states of Ga (2P)and As2 ('2') to the CS0group are 2B2, 2A1,and 2B1. Consequently, fiala~ubramanian~~ argued that the probable candidates for the ground state cf GaAs2 are 2B2,2A1, and 'B1. Table 66 and Figure 36 show the equilibrium geometries and energies of the three electronic states (2B,, *A,, and 2B1).,759The geometries in ref 359 were in slight error due to the ECP parameter mentioned in GaAs work. Corrected geometries are reported here. As seen from Table 66, all three states have isosceles triangular geometries. The 2B2state is the ground state of GAS,. The apex angle of the 2B2state (-47') is the smallest of the apex angles of the three states, implying enhanced As-As bonding over Ga-As bonding, since the Ga-As bonds are the equal sides of the isosceles triangle and the As-As bond corresponds to the base of the isosceles triangle. The As-As bond length for the 2B2 state (MRSDCI) was found to be 2.201 A. The apex
0.916 - , 10; a
From ref 359.
angle of the 2A1state (-58.5') is the largest while that of the 2B2state is the smallest (47'). The MRSDCI calculations for the 2Bzand 2A1states change the Ga-As bond length by only 0.03-0.04 A with respect to the 5320-CASSCF. The apex angle of the isosceles triangle decreased at most by 0.5-0.7' (Cl%). The T,value of the 2A1state obtained by the MRSDCI method is 0.67 eV. Consequently, the theoretical geometries using the 5320-CASSCF were found to be within 1.5% of the MRSDCI results while the energy differences were within 7%. Consequently, Balasubramanian359believed that the 5320-CASSCF method is a more attractive method since it is computationally less intensive. For this reason, Balas~bramanian~' did not carry out MRDCI calculations for the excited 2B1 state. Nevertheless, the 5320-CASSCF geometry for the ,B1 state should be within 2% of the MRSDCI values, and it is believed that the Ga-As bond length would become longer accompanied by a decrease in the As-As bond lengths. As seen from Table 66, the GaAs2trimer is found to be more stable than Ga(2P) As2('Z,'). The dissociation energy for separating the Ga atom from GaAs2was calculated as 1.32 eV at the CI level and 1.11eV at the MCSCF It is anticipated that the experimental De 1 1.5 eV for removing Ga from GaAs2. Note that the theoretical De(Ga2-Ga) is 1.4 eV (cf. section V.A). Consequently, D,(As,-Ga) and D,(Ga,-Ga) are comparable. The Ga-As bond length (MRSDCI) of 2.76 8,in the 2Bz state is comparable to the CASSCF/FOCI bond length of 2.64 8, for the 32-ground state of GaAs obtained by B a l a ~ u b r a m a n i a n .However, ~~~ considering that this bond length is 0.09 8, longer than the experimental value, a corresponding correction should be applied for G A s 2 (see section 1V.A). The As-As bond length of the 2B2state (2.20 8,) is comparable to the As-As bond length of 2.16 8, for the l2+ground state of As2 obtained earlier.159 The As-As tond length of the 2Al state is 2.32 A, which is much larger, implying weakening of the As-As bond in the 2A1 state. The 2B2 state exhibits enhanced As-As bonding and is more stable than the 2A, state, which shows greater Ga-As bonding. This is consistent with earlier theoretical
+
Chemical Reviews, 1990, Vol. 90, NO. 1 151
Heavy p-Block Dimers and Trimers
TABLE 68. Mulliken Population Analysis of the Natural Orbitals of the MRSDCI Wave Functions of the 'Bz and 'A, States of GaAs2'
-___ O t d 1- -__I __-_State
'Al
-
KA?Fl a l i o n - -
Total ----
L 1
As')
Ga(s)
Ga(p) A s ( s )
As(p)
Ga
AS''
2.43
10.23
2.01
0.33
4.11
6.24
2.60
2.22
9.92
1.53
0.56
4.1JG
5.92
2.64
Over1 apC
Gross Loyulation Ga(s) Ga(p) A s ( s )
As(p)
10.40
1.84
0.64
3.75
G.40
0.34
10.35
1.52
0.99
3.73
6.33
0.86
From ref 395. *As population refers to the populations of both As atoms.
e
Overlap population between Ga and both As atoms.
TABLE 69. Geometries and Energy Separations of the calculations (see section 111) that have shown that DeLow-Lying States of Geza (As,) > D,(GaAs) > D,(Ga2). Consequently, among the state re ( A ) Be ( " ) E (kcal/mole) smaller clusters, the states that exhibit greater As-As bonding appear to be more stable. It was predicted359 that the order of stabilities for the trimers of formula A, 2.39 0 Ga,Asy ( x + y = 3) should be AE(As3)> AE(GaAsz)> AE(Ga2As) > AE(Ga3), where AE represents the atom1 2.40 180.0 1.8 ization energy. Balasubramanian's theoretical results 29' on those species and diatomics containing Ga and As 3 2.64 63 1.5 were used by ReentsW in the interpretation of chemical Ah etching of GaAs. OTheoretical constants are from ref 361. Bond distances are asTable 67 shows the leading configurations in the sumed to be the same in these SCF/CI calculations, although for the 'Al state, the base must have different bond lengths compared MRSDCI wave function of the ,B, and ,Al states of to the sides. Thus, the geometries are not fullv oatimized. G ~ A S , . ,As ~ ~seen from Table 67, the leading configurations of these two states are l a ~ 2 a ~ 3 a ~ 4 a q l b ~ 2 b ~ l b q ploying a large McLean-Chandler all-electron Gaussian and la~2aq3a~4a~lbi2bilbq, respectively. basis sets for Si3 and Si3+. Comparable calculations The lal natural orbital of the MRSDCI wave function were also made by Bala~ubramanian,~~ on Si4 using of the ,B, state at its equilibrium geometry was found to be predominantly composed of As 4s while the 2al RECPs. Pacchioni and KouteckP6l found basically two very orbital is mainly Ga 4s but mixes more with As 4s,4p, low lying electronic states for Ge, of 'Al and 3A2/ symand d compared to the Ga-As mixing in the lal orbital. metry analogous to The geometries and energy The 3al orbital is a mixture of bonding As(sl) + Ads,), separation for the electronic states of Ge, are shown in Asl(py)- As2(py)and small amounts of Ga s and Ga pz. Table 69. Balasubramanian's CASSCF/MRSDCI reThe 4al orbital was found to be a mixture of Gab), sults for Si, and Si3+ are shown in Table 70.367 PoGa(p,), As(s) (small amount), and Asl(p,) + As,(p,). tential energy curves for the two lowest electronic states The lbz orbital is predominantly Asl(s)-Asz(s)while the of Si, obtained by B a l a ~ u b r a m a n i a nare ~ ~shown ~ in 2b2 orbital is a mixture of Ga(p,) and Asl(py) + As2(py) Figure 37. (py, + p ) linear combination. The l b l orbital was As seen from the two tables, both Si, and Ge, form (p, + px ) a bonding orbital perpendicular mainly a 'Al ground state with an equilibrium geometry of an to the plane while the 2bl orbital is mixture of Ga(p,) isosceles triangle. The ,B2 state forms an equilateral and As (p + p,J. The coefficients of the d functions triangular structure with a longer Ge-Ge bond length. were f o u d to be nonnegligible for the 2b2, 4al, and 2bl The 'Al-'Z+ separation calculated by Pacchioni and orbitals. K o ~ t e c k y ~ ~ kcal/mol) ~ ( l . 8 appears to be a bit too low Table 68 depicts the net, gross, and overlap Mulliken compared to the corresponding separation of 18 kcal/ populations of the MRSDCI natural orbitals of the ,B2 mol for Si, calculated by Balasubramanian using the and 2A1states of GaAs2. As seen from Table 68, the 2B2 CASSCF/FOCI method.367. state exhibits a greater total net population and less Table 71 shows the absolute energies in hartrees for overlap population (between Ga and the two As atoms) the 'Al and ,B2 states using two sets of d-type polarithan the 2A1state. Thus, the ,A1 state shows enhanced zation functions. The ground state at this level is a 'Al Ga-As bonding compared to the ,Bz state. This is state. The final 1A1-3Az'separation of Si, is 4.6 kcal/ consistent with the earlier qualitative analysis of the mol. The earlier calculations by Grev and S ~ h a e f e r , ~ ~ natural orbitals of the two states. Further, the ,B2 and had shown how sensitive this separation is to the basis ,A, states are ionic (see Table 68). set and electron correlation effects. A SCF/SDCI with as the ground state of Si3, double-f + Id set gave a C. Ge, and Si, which led Grev and Schaefer3= to predict that there are two nearly degenerate isomers for Si3. Pacchioni and K ~ u t e c k y ~have ~ ' investigated the low-lying states of Ge, ( n = 3-7) using a simple Hartree-Fock single-configuration SCF followed by sinD. In, gle-reference CI calculations. Related Si, clusters have also been studied theoretically by many a ~ t h o r s . ~ ~ ~ -Feng ~ ~ and ~ B a l a ~ u b r a m a n i a carried n ~ ~ ~ out complete active space MCSCF followed by multireference singles Calculations at the HF/MP4 level were made by Raghavachari on Si, (n = 3-10).363*364 B a l a s ~ b r a m a n i a n ~ ~ + doubles CI calculations which included up to 177000 configurations on seven low-lying electronic states of made high-level CASSCF/MRSDCI calculations em0
A's
152 Chemical Reviews, 1990, VoI. 90, No. 1
Balasubramanian
TABLE 70. Properties of t h e Low-Lying States of Si3" State
TABLE 71. Properties of t h e 'A, and *B2(:A{) States of Si3 Calculated with Two Sets of d-Type Functions"
E (Hartrees)
Method
,(AIb
9
1410-CASa
2.17
73.6
-856.71138
5410-CAS
2.19
73.6
-866.72846
A1
5410-CAS
A.I
RFOCI
B2
5419-CAS
B2
RFOC I
State
Method
1
RFOCI
2.19
79.6
-866.78532
MRSDCI
2.19
79.6
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Oijkl-CAS stands for a CASSCF calculation in which i a1 orbitals, j bz orbitals, k bl orbitals, and 1 az orbitals were included in the active space. From ref 367. R refers to the two equal distances of the isosceles triangle.
I L
v
I
-1 \ e
F i g u r e 37. Potential energy curves for the 'Al and 3B2states of Si3 (reprinted from ref 367; copyright 1986 Elsevier Science Publishers B.V.).
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" From ref 367. R refers to the two equal distances of the isosceles triangle. TABLE 72. Geometries of Energies of t h e Bent Electronic States of Ina" MRSDCI
CASSCF state
8 (deg) rb)
(A) Ec) (ev) state
8 (deg)
rb)
(F)
Ecj (eV)
4h2
75.2
2.97
0.0
4A2
71.04
2.95
0.0
'B1
54.8
3.25
5.03
4B1
55.2
3.12
0.Gi
281
47.6
3.37
0.07
*B1
48.9
3.31
C.11
2 ~ 1
49.6
3.60
0.28
4 ~ 2
60.0
3.29
0.35
4B2
58.4
3.39
0.35
'A1
51.0
3.49
0.45
a From ref 369. bThe two equal sides of the isosceles triangle; B = apex angle. cZero energy for CASSCF is -5.513007 hartrees; zero energy for MRSDCI is -5.609780 hartrees. The zero energy is for the 'Az ground state at the reported geometry.
They employed a valence (3s3pld) basis set used earlier for In2 together with 3e-RECPs for the In atom. In addition, spin-orbit effects for the low-lying states of In, were obtained. Complete bending potential energy surfaces as a function of bending angle were also obtained.369 Table 72 shows the equilibrium geometries of the bent states of In3 and their energy separations at both the CASSCF and MRSDCI levels of theory. Figure 38 shows the optimized MRSDCI equilibrium geometries of the electronic states of In,. Figure 39 shows the actual bending potential energy surfaces for several electronic states obtained with the CASSCF method. As seen from Table 72, the equilibrium geometries predicted by CASSCF calculations are not substantially different from the MRSDCI geometries. The bond lengths changed at most by 0.13 A. The bond angles changed at most by 4 O at the MRSDCI level of theory compared to the zeroth-order CASSCF level. Table 73 shows the equilibrium geometries and the energy separations of the linear electronic states of In,. MRSDCI calculations predicted two nearly degenerate states of 4A2and 4B1symmetry as candidates for the ground state. The equilibrium geometries of both 4A2and 4B1are isosceles triangles, although 4B1is very acute. Note that the linear limit of the 4A2 (4Zi) state is only 0.07 eV above the ground state.
Chemical Reviews, 1990, Vol. 90, NO. 1 153
Heavy p-Block Dimers and Trimers
TABLE 73. Properties of the Linear Electronic States of Insa MRSOCI
CASSCF State
2 Z"
4A9 22 +
U
'A
2B2
I
4%
Figure 38. Equilibrium geometries of the low-lying states of In3 (reprinted from ref 369; copyright 1989 Elsevier Science Publishers, B.V.).
(A)
E [eV)
State
P,
[A)
i (e\')
-9.02
"r;
2.90
0.07
2.90
0.15
2
3.05
0.37
3.14
0.73
3.13
0.77
2.95
0.57
2.92
0.85
2.85
1.29
2.90
i.13
3.07
4 -
52
R
4% From ref 369.
469 2i+U 4 TTg
TABLE 74. Effect of Including the 4d1° Core in the CASSCF Calculations of the Electronic States of Inla'
4
A2
2 Bl
2
3.01
75.3
2.97
75.2
3.36
49.1
3.37
47.6
3.62
49.7
3.60
49.6
aFrom ref 369.
TABLE 75. Dipole Moments of the Electronic States of In3 with Isosceles Triangular Geometry'
0.0 I 1
0
1 I
I
I
0
20
40
I
60
I
I
SO
100
e
I
I
120 140
I
I
160
IS0
-+(O)
Figure 39. Bending potential energy curves for the low-lying states of Ina (reprinted from ref 369; copyright 1989 Elsevier Science Publishers, B.V.).
The geometry of the 2Bl state is acute and thus the In-In bond lengths for the two sides of the isosceles triangle are large. As seen from Figure 38, both 4B1and 2B1have short In-In bond lengths at the base of the Ti1and , 2A1electronic isosceles triangle. Hence, the 4B1, states exhibited enhanced bonding at the base while for the other states, bond strengths a t the equal sides of the isosceles triangles are greater. Table 74 shows the effect of 4d10 shells on the geometries of the low-lying states of In,. As seen from Table 74, the effect of the 4d1° shell on the bond lengths is
Heavy p-Block Dimers and Trimers
(104) Bond bey, V. E.; Reents, W. D.; Mandich, M. L., unpublishedlresults. ___ - - -..
.
-.
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fl.
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